Markus Vilkki

Flyback transformer of an auxiliary powersupply in photovoltaic inverters

School of Electrical Engineering

Thesis submitted for examination for the degree of Master ofScience in Technology.

Espoo 24.11.2014

Thesis supervisor:

Prof. Jorma Kyyra

Thesis advisor:

M.Sc. Simo Mattila

aalto universityschool of electrical engineering

abstract of themaster’s thesis

Author: Markus Vilkki

Title: Flyback transformer of an auxiliary power supply in photovoltaicinverters

Date: 24.11.2014 Language: English Number of pages: 10+97

Department of Electrical engineering and automation

Professorship: Power electronics Code: S-81

Supervisor: Prof. Jorma Kyyra

Advisor: M.Sc. Simo Mattila

The aim of this thesis was to design flyback transformers for two flyback converters,which are part of an auxiliary power supply of a photovoltaic inverter. In thedesigns, cost efficiency and reliable operation were emphasized and the operationof the designed components was to be verified in a laboratory environment.The cost efficiency in the designs was sought by using triple insulated wire only inthe windings requiring reinforced insulation. In addition, Chinese suppliers wereselected as the manufacturers of the chosen magnetic ferrite cores.Six different prototypes were designed according to the initial preferences and bycalculating parameters and selecting properties. The costs of the different designswere found to be dependent on the size of the component as the largest componentswere the most expensive. In addition, triple insulated wire of multiple strands wasfound to be the most expensive material.The designed properties were found to correspond well with the measured values.In addition, the designed flyback transformers operated without any faults inthe intended flyback converter applications. However, based on the measuredswitching frequencies of one flyback converter, the value of the designed mutualinductance should be somewhat decreased in order to be verified of the reliableoperation.The measured resistances of the windings showed that using multiple individuallyinsulated strands in the wires reduces the increase of the alternating current re-sistance and therefore the amount of copper losses. Furthermore, this was alsoverified as the lower thermal rises of the components were multiple strands wereused, if compared to the equivalent components using single conductors in thewindings.Based on the satisfying operation during the measurements, two designs wereselected for additional testing in the actual photovoltaic inverter application.

Keywords: flyback transformer, design,power supply, winding

aalto-yliopistosahkotekniikan korkeakoulu

diplomityontiivistelma

Tekija: Markus Vilkki

Tyon nimi: Aurinkosahkovaihtosuuntaajan apujanniteteholahteenflyback-muuntaja

Paivamaara: 24.11.2014 Kieli: Englanti Sivumaara: 10+97

Sahkotekniikan laitos

Professuuri: Tehoelektroniikka Koodi: S-81

Valvoja: Prof. Jorma Kyyra

Ohjaaja: M.Sc. Simo Mattila

Taman tyon tavoitteena oli suunnitella flyback-muuntajat kahdelle flyback-teholahteelle, jotka ovat osa aurinkosahkovaihtosuuntaajan apujannitete-holahdetta. Suunnittelussa tuli erityisesti kiinnittaa huomiota komponenttienkustannustehokkuuteen seka toiminnan luotettavuuteen. Suunniteltavien kom-ponenttien toiminnasta tuli myos varmistua laboratoriomittauksin.Kustannustehokkuutta etsittiin mitoituksissa kayttamalla kolmoiseristettya joh-dinta ainoastaan kaameissa, jotka vaativat turvaerotuksen muista piireista sekavalitsemalla magneettisten ferriittisydanten valmistajiksi kiinalaisia vaihtoehtoja.Alkuarvojen, laskettujen parametrien seka valittujen ominaisuuksien perusteellasuunniteltiin kuusi erilaista mitoitusta. Suunniteltujen komponenttien koonhavaittiin vaikuttavan kustannuksiin siten, etta suurimmat komponentit olivatarvokkaimpia. Lisaksi, monisaikeinen kolmoiseristetty kaamilanka osoittautui ar-vokkaimmaksi yksittaiseksi materiaaliksi komponenteissa.Komponenttien suunniteltujen ominaisuuksien havaittiin vastaavan hyvin mi-tattuja arvoja. Suunnitellut komponentit myos toimivat vikaantumatta osanaflyback-teholahteita. Toisen flyback-teholahteen mitattujen kytkentataajuuksienperusteella kuitenkin havaittiin, etta mitoitusten keskinaisinduktanssin tulisi ollajonkinverran pienempi, jotta toiminnan luotettavuudesta pystyttaisiin varmistu-maan kaytannossa.Mitatuista kaamiresistansseista havaittiin useasta saikeesta koostuvan kaamilan-gan pienentavan vaihtovirtaresistanssin kasvua seka siten myos kaamitysten ku-parihavioiden suuruutta. Tasta varmistuttiin mittaamalla alhaisempia lampene-mia komponenteista, joissa kaytettiin useasta saikeesta koostuvaa kaamilankaaverrattuna vastaaviin yksittaisella johtimella toteutettuihin komponentteihin.Tyon tulosten perusteella kaksi mitoitusta valittiin aurinkosahkovaihtosuuntaa-jassa tapahtuviin mahdollisiin jatkomittauksiin.

Avainsanat: flyback-muuntaja, mitoitus,teholahde, kaami

iv

Preface

I want to thank my advisor, Simo Mattila, for the professional guidance during thethesis process. In addition, I would like to thank Professor Jorma Kyyra for thesupervision and the comments.

Special thanks to Joonas Puukko for initiating this process and making it pos-sible. Furthermore, I am thankful for the advices I received from Lari Nousiainenand Jukka Pari regarding the thesis.

Finally, I want to thank my parents, Emma and other friends who have supportedand motivated me during the past eight months and also during my studies in Aalto.

Otaniemi, 24.11.2014

Markus Vilkki

v

Contents

Abstract ii

Abstract (in Finnish) iii

Preface iv

Contents v

Definitions and abbreviations vii

1 Introduction 1

2 Auxiliary power supply of a photovoltaic inverter 22.1 Grid connected photovoltaic inverters . . . . . . . . . . . . . . . . . . 2

2.1.1 Three-phase string inverter application . . . . . . . . . . . . . 22.1.2 Main functions . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.3 Operation of auxiliary power supplies . . . . . . . . . . . . . . 42.1.4 Insulation requirements . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Switched-mode power supplies . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Semiconductor switches . . . . . . . . . . . . . . . . . . . . . 62.2.2 Operating principle . . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Steady state operation of buck-boost topology . . . . . . . . . 132.2.4 Single switch flyback converter . . . . . . . . . . . . . . . . . . 162.2.5 Double ended flyback converter . . . . . . . . . . . . . . . . . 182.2.6 Quasi-resonant control of flyback converters . . . . . . . . . . 20

2.3 Flyback transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Basic theory of magnetics . . . . . . . . . . . . . . . . . . . . 242.3.2 Magnetic core . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.3 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.4 Core size and material . . . . . . . . . . . . . . . . . . . . . . 342.3.5 Winding turns and wires . . . . . . . . . . . . . . . . . . . . . 382.3.6 Losses and thermal rise . . . . . . . . . . . . . . . . . . . . . . 392.3.7 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.8 Manufacturing process . . . . . . . . . . . . . . . . . . . . . . 43

3 Design of flyback transformer 443.1 Initial values and design preferences . . . . . . . . . . . . . . . . . . . 443.2 Turns ratio and inductance . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Core size and flux density . . . . . . . . . . . . . . . . . . . . . . . . 463.4 Numbers of turns, air gap and peak current . . . . . . . . . . . . . . 473.5 Allowed thermal rise and losses . . . . . . . . . . . . . . . . . . . . . 493.6 Creepage margins and arrangement of windings . . . . . . . . . . . . 513.7 Diameters of wires and ohmic losses . . . . . . . . . . . . . . . . . . . 523.8 Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

vi

4 Verifications of flyback transformers 574.1 Measurement of inductances . . . . . . . . . . . . . . . . . . . . . . . 574.2 Measurement of resistances . . . . . . . . . . . . . . . . . . . . . . . . 624.3 Operation in different operating points . . . . . . . . . . . . . . . . . 644.4 Thermal rise with minimum and maximum input voltages . . . . . . 69

5 Summary and conclusions 73

References 75

A Appendix 79

B Appendix 81

C Appendix 84

vii

Definitions and abbreviations

Definitions

αi fraction of window allocated to the windingω angular frequency, 2πfρc resistivity of copperρFe density of core lossesθ phase angle of impedance in degrees∆B peak to peak flux density∆iL change in inductor currentµ characteristic permeability of materialµi initial permeability of materialµ0 permeability of vacuumµa amplitude permeability of materialµr relative permeability of materialΦ flux< reluctance<c reluctance of a core<g reluctance of an air gapA total surface area of a componentAc cross-sectional area of a coreAwi copper area of the winding wireB flux densityBr remanence flux densityBsat saturation flux densityC capacitancecDS drain-source capacitanceD duty ratiodc diameter of coil formerdcp diameter of center poleDmax maximum duty cycle of the MOSFETDres duty cycle of the resonancedwi diameter of a conductordwi,p diameter of a strandE energyEmax maximum energy throughput of a coreF fringing flux factorf frequencyfgrid grid frequencyfsw switching frequencyfsw,max maximum switching frequencyfsw,min minimum switching frequencyfres resonance frequencyFR AC-to-DC resistance ratioG window width of a magnetic core

viii

H strength of a magnetic fieldHsat strength of a magnetic field causing saturation in a coreHg,sat strength of magnetic field causing saturation in gapped coresiC capacitor currenti currentiD drain currentid diode currentIDC DC value of currentiL inductor currentIL DC value of inductor currentIO DC value of output currentii,rms rms value of a current in a windingiM magnetizing currentiP current in primary windingiipk peak value of a current in a windingiPpk peak value of a current in a primary windingir current through range resistorirating maximum rated current of a MOSFETiS current in a secondary windingiSpk peak value of a current in a secondary windingitot sum of rms values of winding currentsix current flowing through DUTJ current densityL inductancelc mean length of a magnetic path in a corele mean length of a magnetic pathlg length of an air gapLl leakage inductanceLM magnetizing inductanceLlP leakage inductance of a primaryLlS leakage inductance of a secondaryMLT mean length per turn of all windingsMLTi mean length per turn of a windingN number of turnsNAUX turns ratio from primary to auxiliary windingNi number of turns in a windingNP number of turns in a primary windingNPS turns ratio from primary to secondaryNS number of turns in a secondary windingt time intervalpi number of parallel strandsPCu copper lossPCu0 ohmic copper loss

ix

PCu,ac AC copper lossPloss total losses of a componentPV power loss densityR0i ohmic resistance of a windingRDS,on resistance of a MOSFET during on timeRr range resistor of the precision impedance analyzerTC Curie temperaturetoff off time of a MOSFETton on time of a MOSFETtres resonance time in discontinuous modetrr reverse recovery timeTCu temperature of windingsTFe temperature of a coreTS switching periodv voltageVAUX voltage over an auxiliary windingVAC,rms rms value of AC voltagevCC operating voltage of control circuitvd diode voltageVD DC value of source voltagevdrop voltage drop due to the RDS,on and a resistance of a primaryvDS drain-source voltageVe effective volume of a corevF forward voltage drop of freewheeling diodevFCC forward voltage drop of diode in auxiliary windingvGS gate-source voltagevin,max maximum input voltagevin,min minimum value of input voltagevL voltage over inductancevO output voltageVO DC value of output voltagevP voltage over primary windingvr reflected voltagevR voltage over a range resistorvrating maximum rated voltage stress of a MOSFETVref reference voltagevS voltage over secondary windingvspike voltage spike due to leakage inductancevx voltage over DUTW width of a coreWA window area of a magnetic coreZ height of a coreZx impedance of a DUTX depth of a coreY thickness of a core

x

Abbreviations

AC alternating currentBCM boundary conduction modeBDEW Bundesverband der Energie- und Wasserwirtschaft,

German association of energy and water industriesBJT bipolar junction transistorCCM continuous conduction modeDC direct currentDCM discontinuous conduction modeDUT device under testDVC decisive voltage classEMI electromagnetic interferenceETD economical transformer designFFM frequency foldback modeIEC International Electrotechnical CommissionIGBT insulated gate bipolar transistorIO input outputMn-Zn manganese-zincMOSFET metal oxide semiconductor field effect transistorMPP maximum power pointn negatively doped regionn− lightly doped negative regionp positively doped regionPCB printed circuit boardPCE power conversion equipmentPV photovoltaicQR quasi-resonantrms root-mean-squareTIW triple insulated wire

1 Introduction

As photovoltaics is establishing position as a significant form of renewable energygeneration, the pressure to improve the cost efficiency of PV (photovoltaic) invertersis growing. Nonetheless, cutting costs in wrong places, in order to improve costefficiency, usually decreases the reliability of the products. Therefore, the designerhas to ensure that the costs of components will not be cut such that the reliableoperation of the products is harmed.

The auxiliary power supply of PV inverter provides power for several tasks in theinverter. Since the power electronic devices, used in switched-mode, became com-mon, the auxiliary power supplies have been made using switched-mode converters.The advantages of switched-mode power supplies over the linear power supplies ishigher efficiency and smaller size.

Flyback converter is a widely used converter topology in the auxiliary powersupplies. The advantages of the topology are the small amount of involved compo-nents and that the flyback transformer provides both energy storage and galvanicseparation between the input and output of the converter. For these reasons, flybackconverter is an economical solution for power supplies requiring low power [1].

The flyback transformer is typically the most expensive component of the con-verter circuit. Because the component has significant impact on the operation andefficiency of the converter, the design of the flyback transformer is worth an effort.

The aim of this thesis is to design flyback transformers for two flyback converters,which are used in the auxiliary power supply of a three phase photovoltaic inverter.The focus of the designs will be cost efficiency and reliable operation in all operatingpoints. The cost efficiency in the designed flyback transformers will be sought byusing TIW (triple insulated wire) only in the windings requiring reinforced insulationto provide protective separation from other windings. In addition, Chinese supplierswill be chosen as the manufacturers of the ferrite cores used in the components.Finally, mutual and leakage inductances, resistances of windings and saturationbehaviour of the components will be measured and the operation of the designs,along with thermal rise, will be verified.

In the second chapter of the thesis, the application of the auxiliary power supplyis described. In addition, the chapter presents the theory of switched-mode powersupplies and flyback transformer. The flyback transformers are designed in the thirdchapter according to the initial preferences of the converters and based on the theorypresented in the second chapter. In the fourth chapter, the operation of the designsis verified in the measurements conducted in a laboratory environment. Finally, thefifth chapter summarizes the most important points of the thesis and conclusionsare made based on the results.

2

2 Auxiliary power supply of a photovoltaic in-

verter

In this chapter, the theory of auxiliary power supplies of a three phase photovoltaicinverter application is presented. In the beginning, the typical functions and require-ments of grid connected string inverters are introduced. Secondly, the operation ofswitched-mode converters is described. Furthermore, the description is focused onflyback topologies, which are widely used in auxiliary power supplies. Finally, themagnetic theory and the design of flyback transformer will be presented.

2.1 Grid connected photovoltaic inverters

Grid connected photovoltaic power plants feed the generated electric power to grid.Their input is connected to PV generators and output is connected to the publicAC (alternating current) grid. The power levels of different applications may varyfrom less than one kW to several MW depending on the physical area of powerplants. Moreover, power levels of new PV power plants are constantly groving atthe moment. In this section, three phase string inverter applications are introduced.Furthermore, the main tasks, operation of auxiliary power supplies and the applica-ble standard for insulation requirements in photovoltaic inverters are presented.

2.1.1 Three-phase string inverter application

Power range of three-phase string inverters is at the moment from few kW to thirtykW. For example, inverter manufacturer SMA offers three-phase string inverters ina power range from five to twenty five kVA [2]. Their input voltage range is typicallyfrom few hundred to thousand volts and the range has to be chosen according tovoltage of PV strings. Furthermore, multistring inverters are equipped with multiplestring inputs to allow connection to multiple parallel PV strings.

Multistring inverters are usually intended to be used on top of buildings havingPV generators mounted on the roof. Depending on the size of covered surface area,power of the PV power plant may require usage of several string inverters or a singlecentral inverter. Central inverters can be used in many applications requiring highpower [3, p. 20]. However, if insolation conditions will not be consistent among allPV strings, multiple PV string MPP(maximum power point) trackers or multipleinverters are used.

A configuration of a typical PV power plant consisting of four panel strings, mul-tistring inverter and grid connection is shown in Figure 1. PV strings are connectedto the inverter by double insulated DC (direct current) cables and small resistanceconnectors [3, p. 47]. DC cabling can be routed directly to string inputs of an in-verter or otherwise through a junction box, if external DC switch or circuit breakersare used. AC cabling consists of three phase conductors in addition to the neutraland the protective earth conductors. AC cabling from inverter to grid is usuallyrouted to AC distribution board of buildings.

3

Inverter PV string

Grid

L1 L2 L3 N PE

Figure 1: Typical three-phase PV power plant configuration.

2.1.2 Main functions

The main task of PV inverters is efficient power conversion from DC to grid-compatibleAC. This can be realized using a power electronic inverter.

In PV inverters, the semiconductor switches are typically IGBTs (Insulated gatebipolar transistors) or MOSFETs (Metal oxide semiconductor field effect transistors)operated in the saturation region during on time. A switch turn on is realized byapplying voltage to the gate of the transistor to make it conduct current from DClink of power converter circuit. Switching moments are timed such that sinusoidalthree phase AC current starts flowing towards grid at grid frequency.

Because the output current of power converter contains high amount of harmon-ics, the current has to be filtered. Third order LCL-filter is commonly used in PVinverters to comply with the standards for injected harmonics [4, p. 31].

Inverters usually operate at efficiencies over 95 percent to maximise the yield ofharvested energy. To be able to maximize the yield, inverters need to constantlytrack the MPP of PV generators. MPP of PV generators depend upon insolationconditions and ambient temperature of the solar panels. In addition to MPP track-ing, inverters collect and store data about the power plant and energy production.

Some grid codes, such as BDEW (Bundesverband der Energie- und Wasser-

4

wirtschaft), determine that PV inverters are not allowed to disconnect from AC gridduring grid disturbances [5]. Furthermore, they are subjected to operate at powerfactors also other than unity when connected to grid. In order to detect changinggrid conditions, inverters monitor connection to grid constantly. Under longtermgrid fault conditions inverter needs to disconnect the PV power plant from grid tomaintain safety of service personnel.

2.1.3 Operation of auxiliary power supplies

Power supplies are generally used in devices to provide power for functionalities suchas described in 2.1.2. Due to a number of critical functions in modern inverters,power supply has to work reliably in all insolation and weather conditions.

Power supply section of grid connected PV inverters may include only a singlepower supply. However, if some functionality is wanted to be maintained duringlong term grid fault and during night time, at least two separately powered powersupplies are needed. The input of the first one will be connected to the DC link ofthe PV inverter and the input voltage range of the power supply will be the sameas the DC link voltage range of the PV inverter.

The other one of the two power supplies is connected to AC grid through a rec-tifier bridge. The rectified grid voltage is filtered to get rid of unwanted ripple ininput DC voltage. Desired low pass characteristics are achieved by adding a filtercapacitor to the output of rectifier bridge. Figure 2 shows the typical voltage wave-forms of full wave rectifier bridge circuit. The operation of the circuit is explainedin detail in [6].

Vo

ltag

e(V

)

Rectified AC voltage

Time(s)

Filtered DC voltage AC grid voltage Average filtered DC voltage

Figure 2: Voltage waveforms of full wave rectifier.

In Figure 2, the maximum value of the filtered DC voltage is calculated by

vin,max =√

2 ∗ VAC,rms (1)

5

where VAC,rms is the rms(root mean square) value of AC grid voltage. From Figure 2and Equation (1) we can see that the maximum value of filtered DC voltage is equalto peak value of AC grid voltage if the forward voltage drop of diodes is neglected.In Figure 2, peak-to-peak ripple in the filtered DC voltage waveform is the voltagedifference between minimum and maximum values. Size of the filter capacitor isselected to meet desired value of voltage ripple.

Typical loads requiring auxiliary power are gate power supplies, grid relays, cool-ing fans, user panel, IO (input output) modules and smaller converters providingpower to measurement circuits and processor. Furthermore, power supplies havemultiple outputs for different tasks. For safety reasons, one separate output is re-quired for functionalities where user has access. These functionalities include forexample user panel, outer fans and IO outputs.

In auxiliary power supplies of PV inverters, output voltages are lower than inputvoltages. Moreover, input voltages of power supplies have typically wide varyingrange but the output voltages are to be regulated to a certain level and are not al-lowed to vary. In order to regulate the output voltages efficiently, feedback controlledswitched-mode converters are used [7, p. 3-5].

2.1.4 Insulation requirements

Requirements for needed insulation in PV inverters are addressed in safety standardIEC (International Electrotechnical Commission) 62109-1 [8]. In the standard, threedifferent DVCs (decisive voltage class) are specified according to different workingvoltage levels. Working voltage is the highest designed voltage occuring when thePCE (power conversion equipment) is operated under worst case combination ofhighest and lowest rated input and output voltages and normal operating conditions.The required insulation level for each circuit is determined by DVCs from Figure A1in Appendix A.

The different insulation levels are functional, basic and reinforced insulation.Functional insulation is the insulation needed only to guarantee normal operation ofa device. In addition to functional insulation, basic insulation provides single level ofprotection against electric shock under fault-free conditions. Reinforced insulationis single insulation system applied to live parts providing equal level of insulationas double insulation. Furthermore, double insulation comprises both basic insula-tion and a supplementary insulation, that provides an additional level of insulationagaints electric shock if basic insulation should fail. However, double and supple-mentary insulations describe how an insulation level is established and they are notconsidered as insulation levels.

In PV inverters, unaccessible circuits require only functional insulation. However,accessible live parts require protective separation or equivalent insulation. Protectiveseparation can be realized for example with reinforced insulation.

The required level of insulation can be realized by using solid insulation or byclearance and creepage distances. Clearance distance means the shortest distancebetween two conductive parts. Similarly, creepage distance means the shortest dis-tance along the surface of insulating material between two conductive parts. More-

6

over in order to determine creepage distances, the pollution degree of the designedinsulator has to be known.

The tables for designing clearance and creepage distances are shown in Figures A3and A4 in Appendix A. In Figure A4, the rms value of the working voltage is usedto determine applicable creepage distances.

When realizing clearances or solid insulation, the impulse withstand voltage andthe temporary overvoltage for grid connected circuits have to be found out from thetable in Figure A2 in Appendix A. The impulse withstand voltage ratings are deter-mined according to applicable system voltages. System voltage for grid connectedcircuits is the rms voltage between phase and artificial neutral point. Furthermore,the system voltage for PV circuits is the maximum rated PV open circuit voltageand they belong in general to overvoltage category ll.

Clearances and solid insulation for circuits connected to grid are designed ac-cording to impulse withstand voltage, temporary overvoltage or working voltage,whichever gives the most severe requirements. For PV circuits, clearances andsolid insulation are determined by working voltage or impulse withstand voltage,whichever gives the most severe requirements.

The typical three phase string inverter application and the main tasks of PVinverters were described in this chapter. In addition, the basic operation of auxiliarypower supplies and requirements for insulations in PV inverters were presented. Inthe following chapter, the main components and the operation of switched modeconverters is analyzed.

2.2 Switched-mode power supplies

Power electronics is widely used in power supplies due to the high efficiency that cantheoretically be close to 100 percent. High efficiency in power electronic converters isachieved by using ideally lossless components such as power semiconductor devicesin switched mode and inductors and capacitors. Further, high efficiency is desirableif the size of the power supply is wanted to be kept small. This is because smallercomponents usually have less cooling surface area compared to a larger componentand the temperature of the smaller mass rises faster. Due to the small amount ofinvolved losses, low temperature rise of a component leads to high power density andsmall overall size. Furthermore, smaller size means smaller manufacturing costs dueto lesser need of materials compared to a bulkier component having lower efficiency.

In this section, the operation of common power electronic converters and semi-conductor switches is described. In addition, the steady state voltage transfer func-tions are derived for a buck-boost converter topology, which is followed by studyingthe operation of two flyback converter topologies and their control in quasi-resonantmode.

2.2.1 Semiconductor switches

The most widely used semiconductors in switched mode power supplies are siliconmade fast-recovery diodes and MOSFETs [6, p. 8]. MOSFETs operated in the

7

saturation region have usually been the most suitable choice for high frequencyapplications rated for voltages less than 500 V [7, p. 81]. The rated voltages ofMOSFETs have been constantly growing and today devices rated for 1500 V can befound from many manufacturers.

The electrical symbol and typical current-voltage characteristics of a diode areshown in Figure 3.

vDiode

(a)

iDiode

vDiodevF

(b)

Reverse

blocking

region

iDiode

iDiode

t

trr

(c)

Figure 3: (a) Electrical symbol, (b) current-voltage characteristics and (c) reverserecovery time of a diode.

Figure 3b presents typical characteristics of p-n(positive-negative) junction diodecurrent id as a function of voltage vd. Figure 3b shows that diode conducts currentonly when forward biased. During conduction, the forward voltage drop vF is usuallyless than one Volt. The value of vF depends on rated blocking voltage and ratedcurrent of the component. Moreover when diode is reverse biased, only a negligiblesmall leakage current passes through it. Furthermore, diodes are rated for certainmaximum reverse blocking voltages until breakdown happens.

Minority carrier devices such as p-n junction diodes and BJTs(Bipolar junctiontransistor) typically exhibit reverse recovery characteristics seen in Figure 3c. InFigure 3c, the current id starts to decrease rapidly in the beginning of turn offtransient. The current decreases below zero to remove stored minority charge fromthe p-n junction [7, p. 76]. When the minority charge is completely removed, thediode can block the reverse voltage and the current id has recovered back to zerowithin time trr. A short reverse recovery time is the reason why fast-recovery diodesare used in high frequency applications such as switched mode converters.

MOSFETs are majority carrier devices exhibiting higher on-state resistance butshorter turn off time compared to minority carrier devices. The electrical symbol ofn-channel MOSFET and the typical current-voltage characteristics are presented inFigure 4.

MOSFETs are turned on by applying voltage vGS on the gate of a MOSFET. Inorder to turn on, the applied voltage have to be higher than the threshold voltage vthof the MOSFET. Figure 4b shows typical current-voltage characteristics with differ-ent applied vGS. When the voltage vGS is increased the drain current iD increases

8

iD

vGS

vDS

Drain

Source

Gate

(a)

vDS

vGS = 10 V

vGS = 4 V

iD

0

(b)

Figure 4: (a) Electrical symbol and (b) current-voltage characteristics of a MOSFET.

rapidly in the saturation region of a MOSFET. A complete MOSFET chip consistsof many parallel cells. The operation of the device can be studied from Figure 5,where the structure of one MOSFET cell is shown.

(a)

n-

p p

GateSource

Drain

n

nn n n p p

GateSource

Drain

n

nn n n

n-

Conducting channel

(b)

Figure 5: (a) Cross section of a power MOSFET cell and (b) conducting channel inon-state.

Typical MOSFET shown in Figure 5a has metallized drain and source pins, whilegate is made of polysilicon. Figures 5a and 5b show the positively and negativelydoped regions inside the device. The region marked by n− is lightly doped and hashigh resistance when the device is in off-state. In the off-state, both p-n and p-n−

junctions of the device are reverse biased and the current does not flow. When thevoltage vGS is applied to the gate, a channel begins to form to positively doped region

9

underneath the gate. When the threshold voltage is exceeded, the resistivity of thedevice has dropped and the MOSFET conducts current iD through the conductingchannel shown in Figure 5b.

The power lost in turn on transient is significant in MOSFETs when vDS isabove 100 V. This is due to drain-source capacitance cDS in which the energy isstored during the switch turn off. The capacitance is shorted every time the switchis turned on and the stored energy in the capacitance is lost. Following equationhas been derived to approximate the lost energy

WCds =2

3cDS(vDS)v2

DS (2)

where cDS is a function of drain-source voltage vDS. Moreover, the square-law de-pendency of WCds from vDS can be noted from Equation (2) [7, p. 98-99].

Semiconductor switches are often considered as ideal components when convertercircuits are analysed. This idealization means that the state of the switch is eitheron or off. Furthermore, transient phenomena, forward voltage drop during on stateand leakage current during off state are neglected.

2.2.2 Operating principle

A circuit to chop input DC voltage to a lower output voltage can be utilized by asingle ideal switch. The resulting circuit and resulting output voltage waveform isshown in Figure 6.

VD VO

switch

(a) (b)

vO

tDTS TS

VD

VDC

Figure 6: (a) A chopper circuit to decrease DC voltage and (b) the resulting outputvoltage waveform.

The switch in Figure 6a can have two different positions. It can connect theoutput vO to the positive terminal of the DC input source VD or alternatively it canshort circuit the load. The resulting steady state output voltage waveform is shownin Figure 6b The value of output voltage vO is VD during the time the switch isconnected to the positive terminal. This time is called the switch conduction timeor on time. Likewise, the time the switch short circuits the load is called the offtime and during that time the output voltage is zero. Furthermore, the switchingperiod TS is known to be inversely dependent on switching frequency TS = 1

fS.

10

By using the circuit shown in Figure 6a, the resulting DC value of the outputvoltage has decreased to a level depending on the switch duty ratio D. The DC com-ponent of the output voltage VO can be calculated by integrating the time dependentvalue of output voltage vO(t) over one switching period TS [6, p. 66]

VO =1

TS

∫ TS

0

vO =1

TS

(∫ ton

0

VD · dt+

∫ TS

ton

0 · dt)

=tonTSVD = DVD (3)

where tON is the on time of the switch, D is the duty ratio and VD is the input sourcevoltage. Although the DC voltage can be decreased using the circuit shown in Fig-ure 6a, the resulting output voltage contains a high amount of harmonics which aremultiples of the switching frequency. The DC component, the fundamental switch-ing frequency and the harmonics, produce the distorted output voltage waveformshown in Figure 6b.

In order to be used in practical applications, the output has to be filtered toget rid of distorted voltage waveform of Figure 6b The desired characteristics arerealized by using LC filter in the output of the converter.

In a practical step-down buck converter, the switch shown in Figure 6a is realizedby using a semiconductor switch, such as MOSFET, in series to the input voltagesource. In addition to the MOSFET, a freewheeling diode is needed to provide pathfor the inductive current present in the true converter circuit during off time of theswitch. The circuit of a buck converter is shown in Figure 7.

VD

VO

iD

LC-filter

L

CD

Figure 7: Step-down buck converter circuit.

The LC filter in Figure 7 consists of inductance L in series and capacitance C inparallel to the load connected to the output of the converter. The corner frequencyof the LC filter is chosen to be lower than the switching frequency to provide desiredlow-pass characteristics [6, p. 66].

In order to analyze the operation of the buck converter in steady state, theoperation of the circuit with idealized components is shown in Figure 8 and Figure 9.

During the switch on time, the switch is considered as short circuit as shown inFigure 8a. The supply voltage VD appears over the diode and the voltage vL over the

11

VO

iL

L

CVD

VL

VO

iL

L

C

VL

(a) (b)

Figure 8: Buck converter:(a) switch on (b) switch off.

inductance becomes VD−VO. The inductor current iL can’t change instantaneouslyand its rate of change follows the well known definition

diLdt

=vLL

(4)

where L is the value of inductance [9, p. 14], [7, p. 17]. Equation (4) can be modifiedto following form to present a change in inductor current diL during a time interval∆t

diL =vL · dtL

(5)

When the switch turns off, the current of the inductor forces the voltage overthe inductor to reverse polarity. The value of vL is now equal to −VO and therectifying diode is forward biased. Therefore, according to Figure 8b, the inductorcurrent iL flows to the output load. The resulting inductor voltage vL and current iLwaveforms are shown in Figure 9. In Figure 9, the waveform of the inductor voltagevL is positive during ton and the inductor current grows according to the Equation(4). During toff , the polarity of vL is negative and the inductor current decreases.The DC level of the inductor current IL is the same as the output current IO of theconverter.

The ripple current of the inductor , shown in Figure 9, flows through the capacitorof the LC filter. This ripple current charges and discharges the capacitance accordingto a definition of capacitors rate of voltage change [10, p. 179]

duCdt

=iCC

(6)

where iC is the capacitor current. Equation (6) shows that the value of outputvoltage ripple depends on the value of capacitance.

Operation of buck converter is limited to only step-down applications. However,if the output voltage is needed to be higher than input voltage, a step up boostconverter can be used. The arrangement of the circuit elements differs slightly frombuck converter. If compared to the buck converter circuit shown earlier in Figure 7,the inductance L has been moved to the input side of the circuit. In addition, thepositions of MOSFET switch and the diode have been moved. The boost converterand equivalent switching circuits are shown in Figure 10.

12

0

0

vL

iL

t

t

ton toff

TS

-VO

VD - VO

IL = IO

Figure 9: Steady state inductor voltage and current waveforms of buck converter.

VD

VO

iL

L

C

D

iC

VO

iL

L

CVD

VL

VO

iL

L

C

VL

(a)

(b)

VD

(c)

iC

Figure 10: (a) Step-up boost converter and the equivalent circuits during (b) switchon and (c) switch off.

During the switch on-time, shown in Figure 10b, the MOSFET switch connectsthe inductance L to the negative terminal of VD. A negative voltage is applied to theanode of the diode and it becomes reverse biased. The inductor current increasesaccording to equation (4) and only the capacitor C supplies energy to output. Theequivalent circuit during switch off-time is shown in Figure 10c. The diode becomes

13

forward biased as the voltage on the anode turns positive. Current starts flowingfrom the inductor to the capacitor, replenishing the drained charge of the capacitor.In addition to inductor, energy is supplied to the capacitor by the input sourceduring the switch off-time [9, p. 32].

2.2.3 Steady state operation of buck-boost topology

A buck-boost converter can be realized by combining buck and boost converters intoa single converter. The advantage of buck-boost converter against either buck orboost topologies is that the output voltage can be lower or higher than the inputvoltage. The steady state output voltage transfer function is the same as the productof transfer functions of the two separate converters [6, p. 81]. Moreover, the outputvoltage polarity of buck-boost topology is reverse. The electrical circuit of buck-boost converter and the operation is presented in Figure 11.

VD

VOiLLC

D

iD

VO

iLLC VOiLL

CVD vL

vL

(a)

(b) (c)

IO IO

IO

Figure 11: (a) Electrical circuit of buck-boost converter and the equivalent circuitsduring (b) switch on and (c) switch off.

Figure 11a show the differences to the two other converter circuits presented inSection 2.2.2. The most notable difference is the position of the inductor betweenthe MOSFET and the negative input terminal. In addition, the diode has beenturned around to conduct current to the opposite direction. The MOSFET stays atthe same position as in the buck converter seen in Figure 7.

When the switch is on, the inductor voltage vL is the input voltage VD as canbe seen from Figure 11b. Conversely, Figure 11c shows that the polarity of vL hasreversed equalling to −VO during the off period. The diode in Figure 11a has becomeforward biased releasing the inductor current to flow to the load.

Cuk [11] and Mohan et al.[6, p. 82-84] have presented the steady state operationof buck-boost converter in different operation modes. In the CCM(continuous con-

14

duction mode), the inductor current never reaches zero remaining continuous at alltimes. In BCM(boundary conduction mode), the inductor current is at the modeboundary. This means that the current reaches zero right at the switch off- andon-time boundary. The inductor current becomes discontinuous in DCM (discon-tinuous conduction mode). This means that the current remains zero for a fraction∆t of switching period. Figure 12 shows the inductor current waveforms in differentconduction modes.

(a) (b)

(c)

DTS (1-D)TS

tOFF Δt

iL

iLiL

t t

t

TS TS

TS

vL vL

vL

(1-D)TSDTS

DTS

IL

Figure 12: Inductor current waveforms: (a) CCM (b) BCM (c) DCM.

Figure 12a shows that when a converter is operated in CCM, the inductor energytransfer is incomplete and the current waveform consists of a peak-to-peak rippleon top of a DC component. The output voltage VO in CCM can be derived as afunction of duty ratio D by equating the integral of inductor voltage vL to zero overone switching period TS:

VDDTS + (−VO)(1−D)TS = 0 (7)

VOTS − VODTS = VDDTs

VO(1−D)

D= VD

VO = VDD

1−D(8)

15

In Figure 12b and Figure 12c, all the energy stored in the inductor is transferredduring one switching cycle. This means that the current is zero and the inductor iscompletely reset. The critical conduction mode, shown in Figure 12b, determines thelimit of operating in CCM and DCM. If the switch on-time is further decreased fromBCM, the current becomes discontinuous and the operation mode changes to DCM.To be noted from Figure 12c, the switching period TS in DCM consists of three timeintervals instead of two in the two other conduction modes. The resonance time tresis added to ton and toff to establish the full period.

The output voltage in DCM can be derived by first calculating the DC level ofinductor current in the current waveform of iL in Figure 12c using Equation (5)

IL =VDDTS

2L(9)

where DTS is the on-time of the switch. Moreover, the DC value of input currentID can be calculated by

ID = DIL (10)

Inserting Equation (9) to (10) yields

ID =VDD

2TS2L

(11)

Using DC values, a relation can be found for input and output powers of the converter

ηPin = Pout (12)

which can be presented also in the following form

ηVDID = VOIO (13)

Inserting Equation (11) to (13), the output voltage VO in DCM is obtained

VO = ηV 2DD

2TS2LIO

= ηV 2Dt

2onfsw

2LIO(14)

where on time is calculated by ton = DTS = Dfsw

.If efficiency of the converter, input voltage, output power and switching frequency

are known, the needed inductance can be calculated by modifying Equation (14) tothe following form

L =(VD − vdrop)2t2onfsw

2(VO + vF )IO(15)

where vdrop is the voltage drop due to the resistance of the MOSFET RDS,on andthe winding R.

By comparing Figure 12a and Figure 12c, we can note that the ripple componentof the two current waveforms is higher in DCM if both waveforms have the same DCcomponent. Therefore according to Equation (5), the inductance value required inDCM is smaller, which usually means a physically smaller inductor. On the other

16

hand, higher current ripple means higher transistor peak currents and a higheroutput capacitor current ripple [10, p. 123].

2.2.4 Single switch flyback converter

As the output voltage of buck-boost converter can be lower or higher than the inputvoltage, the same applies to flyback converter. Flyback converter is based on buck-boost topology but they differ from the magnetic circuit, which in flyback providesgalvanic isolation and turns ratio NPS = NP

NSbetween primary and secondary wind-

ing. The magnetic circuit of flyback converter is often called flyback transformer.However, unlike a typical transformer, current in flyback transformer does not flowin primary and secondary windings at the same time and the transformer behaveslike a coupled inductor. Rightly, the transformer is to be considered as an inductorwhen analyzing the behaviour of the flyback circuit [9, p. 117-120].

Typical flyback circuit resembles the buck-boost circuit of Figure 11a but theinductor is replaced by a coupled inductor. In addition, the ends of secondarywinding have been turned around and the diode is typically drawn to the positive endof the winding. The basic electrical circuit of flyback converter and the equivalentcircuits during ton and toff are shown in Figure 13. Moreover, leakage elements orequivalent resistances of an actual converter are not included in Figure 13.

VD

iP

IO

iC

1:N

VOL

D

C

iS

VD

iP

L VL IO

iC

VOC

iS

VS

(a)

(b) (c)

IO

iC

VO

Figure 13: Flyback converter: (a) Electrical circuit and the equivalent circuits during(b) switch on and (c) switch off.

In Figure 13a, the primary and secondary currents, iP and iS, enter the coupled

17

inductor from the ends of windings marked by black dots. In addition, the placementof the MOSFET, below the inductor, enables a more practical implementation offeedback and control circuitry than in buck-boost topology.

During the MOSFET on-time in Figure 13b, the input voltage VD is applied overthe primary winding of the coupled inductor. The current iP rises linearly to thepeak value iPpk according to Equation (5). During on-time, the freewheeling diodein the secondary remains reverse biased blocking the flow of the secondary currentiS. Furthermore, the output current IO is supplied entirely by the capacitor C andthe charge of the capacitor decreases.

During off-time in Figure 13c, the voltage vS trying to resist a change of the mag-netic flux is induced over the secondary winding. In consequence, the freewheelingdiode becomes forward biased and the current iS begins to flow to the output of theconverter. The current in the coupled inductor decreases from the peak value iSpk

iSpk = N2PS ·

VO · toffL

(16)

where NPS is the turns ratio, VO is the desired output voltage, toff is the off-timeand L is the value of required inductance [1]. Furthermore, the decreasing inductorcurrent replenishes the decreased charge of the output capacitor.

When the MOSFET is turned off, the voltage stress vDS over the MOSFETbecomes to

vDS = VD + vr + vspike (17)

where Vr is the voltage reflected from the secondary winding and the vspike is avoltage spike due to leakage inductance of the flyback transformer. Moreover, thereflected voltage vr in Equation (17) can be calculated as

vr = NPS(VO + vF ) (18)

where vF is the forward voltage drop of the diode. During the turn off transient, thevoltage spike vspike is also reflected to the secondary. Figure 14 presents the typicalcurrent and voltage waveforms of a flyback converter in DCM.

Two resonances can be seen from the typical waveforms of vDS and vS in Fig-ure 14. The first resonance, having higher frequency, occurs between the leakageinductance Ll of the coupled inductor and the parasitic capasitance cDS of theMOSFET. The peak value of this resonance vspike has to be taken into accountwhen calculating the maximum voltage stress of the MOSFET during toff withEquation (17). A Typically used value of the peak has been is vspike = 0.3 · VD [9,p. 130]. The effect of the resonance, caused by the leakage inductance, is decreasedby using dissipative voltage snubbers [12].

The second resonance, occuring when the inductor current has decreased to zero,forms between the inductance LP of primary winding and the parasitic capacitancecDS of the MOSFET. The resonance time equals the discontinuous time ∆t betweentOFF and the next turn on shown in Figure 14.

The frequencies of the resonances can be calculated by

18

IP

IS

VDS

VS

0

0

0

0

VD

VO

tON tOFF tres

t

t

t

t

-VD / N

Figure 14: Waveforms of a flyback converter operated in DCM. The dashed, redcircles highlight the two resonances in the waveform of vDS.

fres =1

2π√LP · cDS

(19)

where L is the inductance of primary winding forming the resonance and the cDS isthe parasitic capasitance between drain and source of the MOSFET [1].

A common problem with flyback converters having wide input voltage ranges isthat the duty ratio in DCM approaches zero as the input voltage approaches theupper limit. Consequently, the MOSFET may not turn on compeletely and reachthe saturation region, which increases the losses of the converter and decreases thereliability of operation. [13]

2.2.5 Double ended flyback converter

If the voltage stress during off-time of the transistor becomes a problem, a doubletransistor flyback circuit, shown in Figure 15, could be used instead of the topology

19

presented in 2.2.4.

VD

IOC

D3

VOLM

Ll

D1 D2

(a)

VDLM IOC

D3

VO

D1

D2

LlVl

Vr

(b)

I1I1

I1

iP

Figure 15: Double ended flyback converter: (a) electrical circuit and (b) equivalentcircuit during MOSFET turn off transient.

The main advantage of using the circuit of Figure 15a is that the voltage stressof the MOSFET is limited to the input voltage VD during off-time. This restrictionis established by clamping the negative terminal of VD through the diode D1 to thedot end of the coupled inductor. The other end of the inductor is clamped to thepositive terminal VD through the diode D2 [10, p. 19].

The inductance of the coupled inductor in Figure 15 is divided to the magnetizingpart LM and the parasitic leakage part Ll. In the double ended flyback circuit, theenergy stored in the leakage inductance during on-time is delivered back to the inputsource during the turn off transient.

Energy stored in an inductor can be calculated by

E =1

2LI2 (20)

where I denotes the peak value of inductor current. From figure 15b we can seethat the current I1 flows through Ll as well as through LM during the turn off

20

transient. Moreover, the voltage over LM is the reflected secondary voltage vr andif the voltage stress over the MOSFET is limited to VD, this leaves the rest of theinput voltage VD − vr over the Ll. In order to maximize the energy transfer fromLM to the output, the time the current I1 flows through Ll should be minimized.

According to Equation (4), the decay time of the inductor current depends on theapplied voltage. Thus setting a lower primary to secondary turns ratio NPS leaveshigher protortion of the input voltage over the leakage inductance which decreasesthe time needed for the current I1 to reset to zero in Ll [9, p. 157-160]. Alternatively,choosing a higher turns ratio slows down the resetting.

The disadvantage of the double ended flyback converter compared to the singleswitch topology is the addition of two clamping diodes D1, D2 and the secondMOSFET. Moreover, choosing the double ended over the single ended topologydoes not give advantage over the problem with MOSFET resulting from wide inputvoltage ranges.

2.2.6 Quasi-resonant control of flyback converters

The output voltage of flyback converters is regulated by closing a feedback loop fromthe output to the control circuit. In the control circuit, the value of output voltageis compared to the voltage reference and the gate control signal of the MOSFET ischanged according to the difference. The gate control signal turns the MOSFET onand off as described in 2.2.1.

In order for the transformer to provide galvanic isolation between primary andsecondaries, the feedback loop has to be isolated from the output. Furthermore, theregulation from the output is typically realized by using an opto-isolator betweenthe controller and error from the voltage reference Vref .

In flybacks operated in DCM, turn on of the MOSFET may occur at any timeduring the resonance between the inductance of primary winding and the parasiticdrain-source capacitance of the MOSFET. The lost energy, during switching, canbe approximated by Equation (2) as described in 2.2.1. From Equation (2) we cannote that the lost energy is dependent on the voltage vDS by square law at the eventof turn on. In order to minimize the switching losses, the turn on should occur atthe valley of the resonance. Figure 16 presents a typical feedback loop in flybackconverters.

The valley of the resonance is used for timing the switching in QR (quasi-resonant) controllers. The QR controller initiates the next turn on of the MOSFETbased on the slope of the voltage of the auxiliary winding seen in Figure 16 [14,p. 5]. In addition to the knowledge of the converter state, the winding providesoperating voltage vCC to the control circuit. The waveform of the voltage in theauxiliary winding equals to reflected voltage vr but is scaled down by a turns ratioNAUX according to

NAUX =NP

N=

vrvCC + vFCC

(21)

where N is the number of turns in the auxiliary winding, vFCC is the forward voltage

21

VD

iP

IO

iC

NPS

VOL

D

C

iS

Control circuit Opto-isolator

VCC

Gate

Out In

Gnd

DCC

Vref

Figure 16: A typical control loop of a flyback converter.

drop of the diode DCC .QR controllers modulate the switching frequency as a function of input voltage

and load current. In addition to switching frequency, some controllers will alsomodulate the primary peak current. Because the QR operation requires zero crossingor slope detection from the waveform of vCC to initiate the switching, the controllerforces converters to operate in DCM. Figure 17 shows the primary winding andthe MOSFET drain-source voltage waveforms in traditional fixed frequency and QRcontrolled flyback converters. [15]

The differences in the waveforms of vDS can be noted from Figure 17. Thetraditional hard switching controller changes the duty ratio in order to achieve de-sired output voltage. At the switch turn on, the voltage vDS can be any value ofthe resonance depending on the discontinuous time ∆t. Instead, for example theUCC28600 QR controller from Texas Instruments modulates switching frequencyand peak current of primary by keeping the duty ratio unchanged in QR region.Figure 18 presents the operation of UCC28600 in which the switching frequency fswand peak current iPpk are modulated as a function of required output power.

The QR region in Figure 18 is the highest on-time the converter can achievewith a given input voltage. In order to keep the duty ratio unchanged, the switchingfrequency has to be between the minimum frequency fsw,min and maximum frequencyfsw,max clamps of the controller. The fsw,max is reached if the input voltage increasesor the load current decreases. As a result, the controller begins to operate in DCMkeeping the switching frequency fixed. [15, p. 11].

As the load current has decreased to approximately 30 percent of the maximum

22

0

VD

t t

VDS VDS

VD

0

tON tOFF tres tON tOFF tres

0

VS VS

0

VO VO

-VD / N -VD / N

Figure 17: Secondary winding and drain-source voltage waveforms of a flyback con-verter controlled in DCM. Traditional hard switching on the left and QR on theright.

value, the controller begins to operate in FFM (frequency foldback mode). Out-put voltage regulation in FFM is achieved by modulating the switching frequencybetween the fmax and fmin clamps as the peak value of primary current is keptconstant. If the load current is approximately below 10 percent of the maximum,the controller operates in green mode. In green mode, bursts of 40 kHz pulses areapplied to the gate of the MOSFET and the average switching frequency becomesless than fmin as can be seen from Figure 18. [15, p. 11, 13.]

Converters timing the turn on of the switch to the valley of the resonance aresaid to be soft switching. Apart from lower losses, another important advantage ofsoft switching is lower conducted and radiated EMI (electromagnetic interference)compared to the typical switching.

The highest on-time in QR is calculated from the energy balance of the converter.The voltage of the secondary winding has the same three time periods during oneswitching cycle as the converter operated in DCM. The switching period TS can beequated

TS = ton + toff + tres (22)

where tres is half of the resonance period and can be calculated by tres = 12fres

.To maintain energy balance in steady state, the integral of the applied inductor

voltage must be zero over one switching period. This means that the volt-secondproducts marked by coloured areas in Figure 17 must be equal [7, p. 20-21]. Equatingthe primary side volt-second products equal, as with the buck-boost converter in

23

VDmax

VDmin

fmax

fmin

PO

fSW

QR

QR

DCM

DCM

FFM

FFM

Green

No load Full load

PO

iPpk

VDmax

VDmin

No load Full load

Figure 18: Switching frequency and peak primary current of a converter usingUCC28600 QR controller. Different operating modes of the controller, QR, DCM,FFM and green, as function of load current and switching frequency with two inputvoltages. [1, p. 9].

Equation 7, the off-time toff can be calculated

VDton = vrtoff = NPS(VO + vF )toff (23)

toff =VDton

NPS(VO + vF )(24)

Inserting Equation (24) to Equation (22), the on-time ton of QR controlled flybackconverter can be solved

ton =NPS(VO + vF )(TS − tres)

(VD − vdrop) +NPS(VO + vF )(25)

If the turns ratio NPS is wanted to be calculated according to the maximum dutycycle of the MOSFET Dmax and the duty cycle of the resonance Dres, Equation 25

24

can be modified to the following form, which includes the voltage drop vdrop due tothe resistance of the MOSFET during on time

Dmax =NPS(VO + vF )(1−Dres)

(VD − vdrop) +NPS(VO + vF )

Dmax(VD − vdrop) +NPS(VO + vF )) = NPS(VO + vF )(1−Dres)

Dmax(VD − vdrop) = NPS(1−Dres −Dmax)(VO + vF )

which leads to the turns ratio of QR controlled flyback converter

NPS =Dmax(VD − vdrop)

(1−Dres −Dmax)(VO + vF )=

Dmax

1−Dmax −Dres

VD − vdropVO + vF

(26)

This section described the operation of switched mode converters and the relatedpower electronic devices. In addition, the quasi-resonant control topology was pre-sented. In the following section, the magnetic circuit of the flyback converter willbe studied.

2.3 Flyback transformer

The important tasks of a flyback transformer are to provide energy storage andcoupling with galvanic isolation to a flyback converter. A typical flyback transformerhas single primary, one or more secondaries and one auxiliary winding. The auxiliarywinding is used for the needs of control circuit, as shown in 2.2.6. A single flybackconverter is often used for powering loads with differing needs by means of differentvoltages or requirements for protection. Consequently, multiple secondaries may beneeded with differing primary to secondary turns ratios.

This section presents the needed theory, construction and parameters to designflyback transformers. Furthermore, the manufacturing process of flyback transform-ers is described.

2.3.1 Basic theory of magnetics

Maxwell equations sum up the experimental laws of electromagnetics. The laws areuseful when designing magnetic components in power electronic converters. Am-pere’s law defines the magnetic field intensity vector ~H in a current carrying con-ductor. According to Ampere’s law, current in a conductor produces a magnetic fieldaround the conductor. Figure 19a demonstrates Ampere’s law in a coil with fourturns. The direction of the magnetic field in a coil can be found with the right-handrule. The vector of the magnetic field intensity points to the direction of the thumb,if the current in the coil flows to the direction pointed by other fingers. Moreover,the law states that if ~H is integrated around a closed loop, the result equals thetotal current passing through the loop. The resulting MMF (magnetomotive force)of the coil can be written as

25

∮l

~H · dl =

∮~J · dS = Ni (27)

where ~H is the magnetic field intensity, ~J is the current density vector, dS is thearea of ~J , N is the number of turns and i is the current in the coil. [16, p. 2.]

H

i

lS

i

(a)

Tota

l flux Φ

v

B

(b) (c)

Induced current i

Applied flux Φ

Induced flux Φi

Closed loop

AC

Figure 19: Demonstrations of Maxwell equations: (a) Ampere’s law, (b) Faraday’slaw and (c) Lenz’s law.

The magnetic field intensity produces magnetic flux density B in a material

B = µ0µrH = µH (28)

where µ0 is the permeability in vacuum, µr is the relative permeability of the ma-terial, µ is a special characteristic permeability of the material and the H is themagnetic field intensity. The value of µ0 is constant equal to 4π*10−7 H/m. [16, p.2-3.]

A total magnetic flux Φ can be found by integrating the magnetic flux densityover a surface S. If the B is uniform and perpendicular to the surface S, the integralcan be written in the following form

Φ =

∫S

~B · dS = BAc (29)

where ~B is the vector of the flux density, B is the magnitude of uniform flux densityand dS is the perpendicular surface equal to the area Ac. [16, p. 1-3.]

According to Faraday’s law, if a total flux which varies in time passes through aloop, a voltage is induced to the loop. The loop in Figure 19b represents a windingin which the voltage v is induced by the total flux Φ. The induced voltage v is givenby the time derivative of the total flux

26

v =dΦ

dt(30)

If the flux is uniformly spread over the internal area Ac of the winding, insertingEquation (29) to Equation (30) yields

v =dΦ

dt= Ac

dB

dt(31)

where Ac is the internal area and B is the uniform flux density. [7, p. 496.]According to Lenz’s law, a flux applied to a closed loop induces a current i to the

loop as shown in Figure 19c. As a result, this current induces a flux Φi to opposethe changes in the applied flux Φ. [7, p. 492-493.]

Analogous to Kirchoff’s law for electrical quantities, the Gauss’s law states thatthe flux Φ going in to a surface must equal the flux passing through the surface in aclosed magnetic path. This relation applies to all closed magnetic circuits and it canbe modeled the same as Kirchoff’s law for current in different nodes. The Gauss’slaw is mathematically given by ∮

S

~B · dS = 0 (32)

where B is the flux density and the dS is the surface the flux passes through.Equation (32) states that the net flux through a surface must be zero. [7, p. 499.]

2.3.2 Magnetic core

In Figure 20a, a coil with an air core emits the magnetic field out to surroundings.However, if the coil is spooled on a magnetic core with the permeability µr 1, apath of low magnetic resistance is provided for the flux and the magnetic field staysalmost entirely inside the core. Furthermore, the magnetic path length le can beaccurately determinated which could otherwise be difficult. [17, p. 6-7.] Figure 20bpresents a winding spooled around the magnetic core.

Because the current i in Figure 20 is related to the magnitude of the magneticfield H by Ampere’s law, the MMF of the coil in Figure 20b is

Fm = Hle = Ni (33)

where Fm denotes the MMF, H is the magnitude of the magnetic field, le is themean length of the magnetic path, N is the number of turns in the coil and the i isthe current flowing in the coil. Consequently, the magnitude of the field H can bewritten as

H =Ni

le(34)

and insertion to Equation (28) yields the flux density in the core

27

(a) (b)

H

i

le

Φ

B

vv

i

H

Φ

Relative permeability of the core μr >> 1Relative permeability of the core μr = 1

Figure 20: (a) A coil with an air core and (b) a coil spooled around a magnetic core.

B = µNi

le(35)

where µ is the characteristic permeability of the core.B-H loop presents flux density B as a function of magnetic field strength H of

core materials. Typical B-H characteristics of different cores are shown in Figure 21.In Figure 21a, the slope of the air core is throughout linearly proportional to

µ0 and the B-H loop is a straight line crossing the axles of B and H at the origin.Consequently, the magnetizing path is the same as the demagnetizing path and thecore has demagnetized completely when the magnetic field is zero. On the otherhand, a typical B-H loop of a magnetic core is as shown in Figure 21b. From theloop, a number of characteristics can be observed. The first one is that the slope isnot linear throughout as the magnetic core saturates after the magnetic field strengthH exceeds Hsat or becomes lower than −Hsat.

As the core saturates, the strength of magnetic field H increases but the perme-ability of the core decreases rapidly and the operation of the core begins to resemblethe operation of the air core. The reason for the decrease of permeability is the factthat the flux density B in the core cannot increase any further as it has reached thesaturation point specific to the material. The effect of saturation is not permanentas the core recovers from saturation if the magnetic field decreases to a value belowHsat. In Figure 21b, the direction of the magnetization process is marked by arrowsand the initial magnetization path, marked by the dashed line, starts from zero.

The second characteristic results from hysteresis in magnetic materials. At thepoint of the B-H loop where the magnetic field has decreased to zero a remanenceflux is still present in the core and the corresponding flux density is denoted by Br.The flux is completely reset to zero by a negative magnetic field denoted by coercive

28

B (T)

H (A/m)

B (T)

H (A/m)

Br

Bsat

Hsat

B = μ0H

HC

(a) (b)

B (T)

H (A/m)

Bsat

Hsat Hg,sat

Effect of gapping

(c)

-Hsat

-Bsat

Figure 21: B-H characteristics: (a) air core, (b) a core with µr 1 and (c) effect ofgapping a magnetic core.

field strength HC . [17, p. 10.]The third characteristic is the slope of the linear section of the B-H loop. Trans-

formers and inductors are usually designed to operate in this area and they arenot allowed to saturate under normal operating conditions. The slope of the curvedepends on the characteristic permeability of the core according to Equation (28).

Air gaps are used in inductors to provide energy storage and to increase themaximum allowed magnetic field strength Hsat. The effect of using an air gap in amagnetic core is shown in Figure 21c. The air gap allows the core to be operated withhigher field strengths if compared to an ungapped core by moving the saturation fieldstrength Hsat to Hg,sat. Accordingly, gapping the core decreases the permeability ofthe core and therefore affects the ability of the core to resist flux. This can be notedby analyzing the analogies between electric and magnetic circuits.

Voltage source, current and resistance with electric circuits are analogous toMMF, flux and reluctance with magnetic circuits [7, p. 498]. Nevertheless, shouldbe remembered that resistance is dissipative whereas reluctance is a lossless prop-

29

erty. The equation for MMF, comprising reluctance < and flux Φ, can be found bysubstituting B from Equation (29) into Equation (28) and further substituting theresulting H from Equation (28) into Equation (33)

Fm =leµAc

Φ (36)

where the reluctance < is

< =leµAc

(37)

where le is the length of the magnetic path, µ is the permeability of the core and theAc is the cross-sectional area of the core. By comparing the two latest equations,we can note that the MMF is

Fm = <Φ (38)

where < is the reluctance and Φ is the flux.Equation (37) shows that reluctance depends on the characteristic permeability

µ. Cores with air gaps include two different permeabilities, which are the perme-ability of the magnetic material and the permeability of the air. Since the sameflux passes through the magnetic core and the air gap, the reluctance of the core iscomprised from the reluctance of the magnetic core <c and the reluctance of the airgap <g. The resulting MMF of gapped cores can be written as

Fm = Ni = (<c + <g)Φ = (lcµAc

+lg

µoAc)Φ (39)

where lc is the length of the magnetic path in the core, lg is the length of the airgap and the Ac is the cross sectional area of the leg the winding is spooled on. Sincethe relative permeability of a magnetic core is µr 1, the reluctance of the core issignificantly smaller than the reluctance of air gap. Figure 22 shows a coil spooledaround two different gapped core shapes.

A typical inductor can be made using a C core shown in Figure 22a. However,the generated EMI and noise may produce a problem as the flux fringes out of thecore because of the air gap in the opposite leg. This can be decreased by locatingthe winding on top of the gap, which forces all the flux to pass through the coil.An example of such inductor is shown in Figure 22b. The shown ETD (economicaltransformer design) cores are widely used in coupled inductors, in addition to otherE-shapes, because they have wide window areas around the center pole. The widewindow areas are necessary to minimize the number of layers in coupled inductorshaving multiple windings. [18, p. 3, 5.] The full flux Φ1 in the center pole dividesbetween other legs of the core according to the cross sectional areas of the legs. Themean lengths of magnetic paths as well as the dimensions of cores are reported bymanufacturers of cores in their datasheets.

As described in 2.2.4 flyback transformers are coupled inductors having one pri-mary and at least one secondary winding. Typically all windings are located on topof each other on the same leg of the core. This produces best coupling between the

30

lg

lc

H

i

B

Φ

v

H

Φ

lg

(a) (b)

v

BΦ1

lc

Φ2 Φ3

Φ3 Φ2

Φ1

i

G

Figure 22: (a) Gapped C core and (b) gapped ETD core with the width of thewindow G.

different windings. Figure 23 shows a coupled inductor with a primary and a singlesecondary winding spooled around the center pole of an ETD core.

Φ

vP

BΦ

vS

iSiP

Figure 23: A coupled inductor using ETD core.

During on time, the current iP in Figure 23 flows through the primary windingand the energy is stored in the air gap in the center pole. When the switch is turnedoff, the current iS starts flowing through the secondary diode and the energy storedin the inductor is released to the output of the converter as described in 2.2.3.

31

Voltages and currents in the different windings can be solved by Faraday’s law.Since the same flux links all windings located on the center pole of the core, voltagesover windings can be stated as a voltage over one loop multiplied by the number ofturns in the winding. Therefore the voltages in Figure 23 can be written as

vP = NPdΦ

dt(40)

vS = NSdΦ

dt

and if the flux is eliminated, the voltage ratio, identical to a transformer, is achieved

vP =NP

NS

vS = NPSvS (41)

where vP is the voltage over primary, NP is the number of turns in primary, vS is thevoltage over secondary, NS is the number of turns in secondary and NPS is the turnsratio from primary to secondary. Furthermore, currents in different windings areideally inversely proportional to the turns ratio between primary and the secondaryNPS

iP =1

NPS

iS (42)

where iP is the current in primary and iS is the current in secondary. [7, p. 502.]

2.3.3 Inductance

Voltage over a winding, such as the primary in Figure 23, can also be expressedby means of flux density. Since the same flux passes through all loops and voltageover one loop can be found by Equation (30), the voltage over the winding can beexpressed as

v = NACdB

dt(43)

where N is the number of turns in the winding, AC is the cross-sectional area of the

core and dBdt is the rate of change in the flux density. By inserting Equation (35) to

Equation (43), the voltage over the winding becomes

v =µN2ACle

di

dt(44)

which is of the form

v =N2

<di

dt(45)

and since the reluctance of the core is comprised from both reluctance of the coreand the reluctance of the air gap, Equation (45) can be written as

32

v =N2

<c + <gdi

dt(46)

where <c is the reluctance of the core, the <g is the reluctance of the air gap and didt

is the slope of the winding current. By comparing Equation (46) to Equation (4),we can note that the inductance L of an inductor can be expressed as

L =N2

<c + <g=µ0ACN

2 ∗ 10−2

lcµr

+ lg(47)

where N is the number of turns in the winding, µr is the relative permeability of thecore, lc is the mean length of magnetic path in the core in cm, µ0 is the permeabilityof air and lg is the length of the air gap in cm. Equation (47) shows that increasingthe length of the air gap decreases the value of inductance. On the other hand,the air gap balances the operation of the inductor as the value of inductance is notmerely dependent on the permeability of the core which varies along temperature[7, p. 500].

By decreasing the inductance, the change of the current increases, which can benoted from Equation (46). Therefore, because Equation (20) states that the storedenergy is dependent from the current by square law and directly from inductance,the energy stored in the inductor increases as the length of the air gap increases.Accordingly, this means that the energy of the inductor is mainly stored in the airgap. In addition, the air gap balances the operation of the inductor as the valueof inductance is not merely dependent on the permeability of the core which forexample varies along temperature [7, p. 500].

Some of the flux in practical transformers and coupled inductors does not link allwindings. This incomplete coupling between windings leads to the practical modelof the flyback transformer, where the inductance of the inductor is divided intomutual and leakage parts. The model of the flyback transformer with one secondarywinding is shown in Figure 24.

LM

LlP NPS = NP : NS

vP vS

iP

iM

iSLlS

iS /NPS

Figure 24: Model of a flyback transformer with two windings. [16, p. 26]

33

In Figure 24, leakage inductances LlP and LlS are in series to the relating wind-ings. Furthermore, the magnetizing inductance LM is in parallel to the primarywinding and the magnetizing current iM , flowing through LM , is the sum of primarycurrent iP and secondary current iS. During toff the iP is zero and the current inthe secondary winding iS can be calculated by

iS = NPSiM =NP

NS

iM (48)

where iM is the current flowing through the magnetizing inductance, NPS is theprimary to sescondary turns ratio, NP is the number of turns in primary windingand NS is the number of turns in secondary winding. Therefore, the iM of coupledinductors with n number of secondaries, can be expressed as

iM = iP +n∑i=1

iSiNPi

(49)

where NPi is the turns ratio from primary to the equivalent secondary. Moreover,self inductance LS is the sum of the leakage Ll and the magnetizing inductances LMrelated to the winding. [16, p. 25-26.]

In flyback converters, leakage inductance has the negative effects described 2.2.4and 2.2.5. In addition, leakage inductance has effect on cross regulation betweendifferent outputs. Cross regulation means the coupling of the voltages betweenmultiple outputs. Typically one output of a coupled inductor is regulated and thevoltages of other windings vary depending on the coupling between the secondaries.As a consequence, the coupling between different outputs should by minimized inorder to reduce the cross regulation. Conversely, the leakage inductance relatingto the primary should be minimized to achieve best coupling between the primaryand the secondaries. [20] The leakage inductance caused by bowing of the windingscan be minimized by using a core with a round center pole. This ensures the mostcompact design of the windings. [17, p. 138.]

Due to the presence of an air gap in inductors, the flux in the air gap fringesoutside the air gap. The fringing effect near an air gap is shown in Figure 25. InFigure 25 the winding is spooled on top of a coil former. Fringing flux induceseddy currents to the windings near the gap which causes localized heating of thecomponent. [17, p. 23-24.] The effect of fringing flux to the copper losses has beenstudied and the fringing flux will increase the current density in the wires becauseof the induced eddy currents [21].

The fringing flux reduces the effective length of the air gap decreasing the reluc-tance of the inductor. According to Equation (47) the inductance of the inductorincreases as the reluctance decreases. The effect of fringing flux to the inductancecan be approximated by fringing flux factor F

F =

(1 +

lg√AC

ln2G

lg

)(50)

where lg is the length of the air gap, AC is the cross sectional area of the core and G

34

Core

Core

WindingCoil former

Fringing fluxAir gap

Figure 25: Fringing flux near an air gap. [17, p. 26]

is the width of the window. [17, p. 25.] The correct value of magnetizing inductanceLM , taking into account the fringing, can be calculated by

LM = FL (51)

2.3.4 Core size and material

The needed size and shape of the core can be approximated if the output powerof the flyback converter is known. Manufacturers of cores have typically tables andgraphs for selecting the initial size and shape. For a given power output, the suitablecores from Ferroxcube are presented in Table 1.

Listings, such as Table 1, give an initial idea of usable sizes of different cores.In addition, manufacturers typically have graphs for selecting a core and an airgap for a given stored energy. After the required energy has been calculated, thesize of the core can be selected from the graph provided by manufacturers. Anexample graph for ETD shapes is shown in Figure 26. In Figure 26 the maximumstored energy is plotted as function of air gap length for different sizes of ETD cores.Moreover, the required amount of energy, over one switching period, is determinedby Equation (20).

In the literature, several methods have been proposed for approximating therequired size of the core for transformers [7], [16], [17], [18]. The method for coupledinductors produces a parameter Kg for selecting the required size of a ferrite core.The parameter is given by the geometry of the core

35

Table 1: Power throughput for different types of cores at 100 kHz switching fre-quency. [19, p. 29]

POWERRANGE (W)

CORE TYPE

< 5 RM4; P11/7; T14; EF13; U105 to 10 RM5; P14/810 to 20 RM6; E20; P18/11; T23; U15;

EFD1520 to 50 RM8; P22/13; U20; RM10;

ETD29; E25; T26/10; EFD2050 to 100 ETD29; ETD34; EC35; EC41;

RM12; P30/19; T26/20; EFD25100 to 200 ETD34; ETD39; ETD44; EC41;

EC52; RM14; P36/22; E30; T58;U25; U30; E42; EFD30

200 to 500 ETD44; ETD 49; E55; EC52;E42; P42/29; U67

> 500 E65; EC70; U93; U100; P66/56;PM87; PM114; T140

Kg ≥A2CWA

(MLT )(52)

where AC is the cross-sectional area of the core, WA is the area of the window andMLT is the mean length of a turn. On the other hand, the size of the core can bestated to be a function of design specifications

Kg ≥ρcL

2M i

2toti

2M,max

B2maxPCu0Ku

1010 (53)

where ρc is the effective resistivity of copper, LM is the desired magnetizing induc-tance, itot is the sum of rms currents in all windings referred to primary, iM,max isthe peak value of magnetizing current, Bmax is the maximum allowed flux density inthe core, PCu0 is the value of allowed ohmic copper losses and Ku is the filling factorof the window. The resistivity of copper is 1.724*10−8 Ω-cm at the temperature of25C and 2.3*10−8 Ω-cm at 100C. [7, p. 550-553.] The allowed copper losses PCuare approximated according to the allowable thermal rise of the component.

The filling factor Ku means the percentual area of the window occupied bycopper. The rest of the window has to be reserved for coil former, safety margins,wrapper insulation, layer insulation, wire insulation and the fill factor of the wire.The value of Ku is usually less than 0.3 depending on the chosen wire type. [17, p.132, 142.]

The rms values of winding currents can be calculated similarly as the rms valuesof triangular waveforms

36

Figure 26: Ferroxcube: I2L graph for selecting size from ETD cores. [19, p. 35]

ii,rms = iipk

√D

3(54)

where iipk is the peak value of the current in the winding and D is the duty ratioleading to the peak value. The sum of rms currents itot is calculated with respect toprimary winding

itot = iP,rms +N2

NP

iS1,rms . . .+Nn

NP

in,rms (55)

where iP,rms, iS1,rms and in,rms are the rms values of currents in the primary, the firstsecondary and the secondary number n. In addition, the currents in Equation (55)are multiplied by the equivalent turns ratios from each winding to primary. [7, p.557-558.]

When the parameter Kg has been calculated with Equation (53), the size of thecore can be selected from Table 2. In Table 2 the dcp is the diameter of the centerpole. The parameter Kg of the selected size in Table 2 should be greater than thevalue calculated by Equation (53). This helps fitting the windings, producing the

37

copper losses PCu, to the window of the core and that a practical air gap can berealized to avoid premature saturation.

Table 2: Geometrical constants of different of ETD cores. [7, p. 866.], [19, p.541,545,551,554.]

Core type Kg (cm5) AC (cm2) WA (cm2) dcp (cm) G (cm) lc (cm)ETD29 0.0978 0.76 0.903 0.98 2.2 7.20ETD34 0.193 0.97 1.23 1.11 2.36 7.86ETD39 0.397 1.25 1.74 1.28 2.84 9.22ETD44 0.846 1.74 2.13 1.52 3.22 10.3

Mn-Zn (Manganese-Zinc) ferrite is a widely used material in the cores of flybacktransformers. Due to the low power loss density with typical flux densities andswitching frequencies around 100 kHz, many Mn-Zn materials are suitable for powertransformer applications including flyback transformers [19, p. 19]. Other importantparameters of magnetic materials are for example the BH characteristics, the ampli-tude permeability µa and Curie temperature TC . The saturation flux density Bsat ofthe material can be checked from the BH loop or the amplitude permeability curveprovided by the manufacturers. The curie temperature expresses the temperatureat which the magnetic properties of the core will cease to exist. Hence, the curietemperature of the material should never be reached during normal operation.

A cross-reference listing of Mn-Zn materials presenting complementary materi-als between manufacturers is shown in Figure 27. From the listing in Figure 27,the correspondence of different materials can be noticed between the manufacturers.Ferroxcube states that materials 3C30 and 3C34 are suitable for line output trans-formers and flyback converters operated with switching frequencies up to 200 kHzand 300 kHz respectively. In addition, materials 3C90 and 3C94 are suitable forgeneral industrial use with the same corresponding switching frequency ranges. [19,p. 29.] The listing in Figure 27 shows that for example Asian manufacturers TDGand DMEGC have equivalent materials to 3C30, 3C34, 3C90 and 3C94 from theEuropean Ferroxcube.

Listings, such as Figure 27, should not be trusted exclusively. Instead, the opera-tion and parameters of the materials are to be verified from the datasheets providedby the manufacturers of the materials and by verifying the operation of the com-ponents. The datasheets of Mn-Zn ferrite material 3C30 from Ferroxcube and theequivalent materials from DMEGC and TDG can be found out from Appendix B.

In addition to ferrites, powdered materials are sometimes used in flyback trans-formers operated with high DC flux density. The advantage of the powdered coresagainst typical ferrites is that the air gap is distributed evenly into the core insteadof discrete air gaps. As a result, the external field near the air gap will be dimin-ished. The disadvantage limiting the usage of powdered cores is the higher powerloss if compared to ferrites. [18, p. 5.]

38

Figure 27: Cross-reference list of Mn-Zn ferrite materials. [22]

2.3.5 Winding turns and wires

The minimum number of turns in a primary winding can be determined by modifyingEquation (43) to the following form

NP =vDtonBmaxAC

(56)

where ton is on-time of the MOSFET, vD is the input voltage applied to the pri-mary, Bmax is the maximum allowed flux density swing of the core material and ACis the cross-sectional area of the selected core. From Equation (56) we can note thatchoosing a higher number of turns than calculated leads to a lower flux density. Al-ternatively if the number of turns is chosen to be less than calculated, the maximumflux density will be exceeded. In addition, the chosen value should be integer.

In order to realize the desired turns ratio NPS, the number of turns in the sec-ondaries is calculated by

NS =NP

NPS

(57)

where NS is chosen to be an integer.To minimize the copper losses PCu the area of the window WA should be allocated

to each winding according to their rms currents. Therefore, a fraction of the windowis calculated according to

αi =Niii,rmsNP itot

(58)

where αi is the fraction allocated to the winding, Ni is the number of turns in thewinding, ii,rms is the rms value of the current in the winding and itot is the sum ofrms currents of all windings. [7, p. 547-548.]

39

The copper area of each winding wire Awi can then be calculated by

Awi ≤αiKuWA

Ni

(59)

where Ku is the filling factor of the window, Wa is the area of the window and Ni isthe number of turns in the winding.

The diameter of the round wire dwi can be calculated from the wire area byapproximating the area of the conductor with the area of circle

dwi = 2

√Awiπ

(60)

If multiple parallel conductors are used, the diameter of conductors can be foundby equating the area Awi and the area of parallel strands Awi,p equal

Awi = Awi,p (61)

piπ

(dwi,p

2

)2

= π

(dwi2

)2

and solving the diameter

dwi,p =dwi√pi

(62)

where dwi,p is the diameter of a single strand of pi parallel conductors.Litz wire consists of multiple parallel and individually insulated strands twisted

together. If Litz wire is used, the diameter of a single strand should be smaller thancalculated by Equation (62) because the fill factor of multiple, individually insulatedand twisted, strands is worse than that of the single conductor [16, p. 146].

The current density in the windings can be calculated by

J =ii,rmsAwi

(63)

where ii,rms is the rms value of the winding current and Awi is the copper area ofthe winding. A typical value of the maximum current density is 5 A/mm2 in thecopper wire used in the magnetics of power electronics [26].

2.3.6 Losses and thermal rise

Losses are produced both in the core and in the windings of the flyback transformer.The core loss PFe of ferrites can be divided into two importants parts, hysteresisloss and eddy current loss. Hysteresis loss of the ferrite core depends on the peakflux swing of the B-H loop. The energy lost during one cycle is the volume of thecore multiplied by the area of the B-H loop. The hysteresis power loss is obtained ifthe lost energy during one switching cycle is multiplied by the switching frequency.[7, p. 506.] Alternatively, losses are generated by fast changing magnetic fields in

40

semiconducting ferrite cores. The magnetic fields induce eddy currents to the corewhich produces losses in the resistance of the core material. Furthermore, eddycurrent losses are dependent from frequency by second power. [25]

Core losses are approximated from the graphs in the datasheets presenting thecore loss density as a function of peak ac flux density. The core loss density graphof the ferrite material 3C30 from Ferroxcube is presented in the bottom right ofFigure B1 in Appendix B. From the graph, the core loss density of the ferrite materialcan be examined with the given operating frequencies. Thereafter, the density ismultiplied by the effective volume Ve of the selected core to obtain the core loss PFe.

Copper losses of windings PCu is the sum of ohmic and AC losses. Ohmic lossesare determined according to

PCu0 =n∑i

R0ii2i,rms (64)

where R0i is the ohmic resistance of the wire and ii,rms is the rms value of the currentin the winding. The ohmic resistance R0i of the winding can be calculated by

R0i = ρc4(MLTi)N

pi ∗ πd2wi,p

(65)

where ρc is the effective resistivity of copper, MLTi is the mean length per turn, Nis the number of turns in the winding, pi is the number of strands and dwi,p is thediameter of one strand. [16, p. 42.]

The part of AC losses in flyback transformers is significant because of the typicalswitching frequencies in the range of 40kHz to several MHz. The triangluar windingcurrent is the sum of harmonics of the switching frequency. Therefore, the AC lossis defined by

PCu,ac = FRPCu0 (66)

where FR is the ac-to-dc resistance ratio which takes into account the harmonics inthe current. FR includes the skin effect and the proximity effects of adjacent layersand the fringing flux. [26]

Skin effect means that rapidly changing current in a conductor induces a fluxthat induces eddy currents to the conductor opposing the main current in the centerof the conductor. Conversely, the induced eddy currents increase the current densityin the outer surface of the conductor. This leads to a penetration depth where thecurrent flows in a conductor. The penetration depth in a copper conductor can beexpressed as

δ =

√2ρcωµ

=

√ρc

πfµ0

(67)

where ρc is the resistivity of copper, ω is the angular frequency, f is the frequencyof the current and µ is the permeability of the material which is roughly the sameas the permeability of vacuum µ0 [27].

41

The proximity of the air gap has significant effect to the resistance of the windingsclosest to the air gap. The effects of the induced eddy currents depend on themagnetomotive force, operating frequency, length of the air gap and the distancebetween the gapped leg and the layer of the winding. [21] Windings in inductorsexperience an increased power loss caused by the proximity effect of adjacent layers.In the proximity effect, layers of windings induce flux to the spaces between theadjacent layers. This opposing flux increases the AC resistance of windings byinducing eddy currents to the surfaces of the conductors in the adjacent layers. [7,p. 510–511.]

The effects of eddy currents in conductors can be reduced by using multiple par-allel conductors or bunched wires, such as Litz wire, instead of a single conductor[28]. The diameters of strands are calculated for example according to the pene-tration depth at a certain frequency. In addition, the number of parallel strands isdefined from Equation (62) to match the equivalent area and diameter of the singleconductor specified by Equation (59) and Equation (60).

Losses contribute to the efficiency and the temperature rise of the component.Typically the amount of allowed losses is determined according to the allowed tem-perature rise. The maximum thermal rise and fluctuations in the operating temper-ature are known to affect the life cycle of the component. Accordingly, the higherthe temperature rise and fluctuations, the shorter the life cylce. The operating tem-perature of the component is the sum of ambient temperature and the temperaturerise of the component [16, p. 253].

The three heat transfer mechanisms are conduction, convection and radiation.All mechanisms are dependent on the material and the surface area of the compo-nent. If the allowed temperature rise ∆T and the total surface of the component areknown, the power losses in the component can be approximated with an empiricalequation

Ploss = (∆T )1.1A (68)

where ∆T is the difference between the hot spot and the ambient temperaturesand A is the total surface of the component. Equation (68) yields an accuracy ofaround ten percent and it can be used to predict the dependence of the different coresizes from the allowed losses. Alternatively, methods separating the effects of theheat transfer mechanisms and using the lumped thermal resistances of the structureshould be used with applications requiring better accuracy. [16, p. 256-276.]

2.3.7 Structure

The arrangement of the windings to the window of the core has an effect on the re-sistance, the leakage inductance and the produced EMI of the flyback transformer.The ohmic resistance is dependent on the length of the winding according to Equa-tion (65). Winding with the highest voltage should be located closest to the coreand be connected to the drain pin of the MOSFET. This minimizes the ohmic re-sistance of the winding and enables a shielding effect of other windings decreasingthe radiated EMI. [29]

42

On the other hand, the amount of AC losses resulting from fringing flux near theair gap is significant in flyback transformers. In order to reduce the effect of fringingflux on the windings closest to the gap, the windings should not be located directlyon top of the core. The higher the length of the air gap, the higher distance shouldbe kept to the gap. Nevertheless, typically windings are spooled on coil formersproviding greater than 0.5 mm distance to the gap depending on the size of thecoil former. This distance does not minimize the effect of fringing flux to the ACresistance but is usually the practical choice if multiple windings are needed to befitted to the window of the core. [21]

The winding order has an affect to the leakage inductances and therefore, tothe coupling between windings. If the primary should be closest to the core, asdescribed above, the secondary winding, conducting the highest current, should beon top of the primary. This maximizes the energy transfer between windings asthe leakage inductance is proportional to the distance between the windings. If theprimary consists of multiple layers, interleaving the secondaries between the layersof the primary decreases the leakage inductance approximately by half. Moreover,the total leakage inductance of the component should be minimized because leakageenergy between any of primary to secondary combinations has an effect on theefficiency of the converter. [21]

Different layers should be spread to the entire allowed width to maintain thegood coupling. Therefore, coil formers with separate sections next to each othershould be avoided with flyback transformers. [30] A method for spreading a layerwith only a couple of turns is to use parallel wires having equal wire area to theoriginal single wire. An interleaved transformer design and the usage of parallelwires is presented in Figure 28.

First half of primary First secondary Second secondary Second half of primary

Equal area

Margin tape Insulation tape Coil former

Figure 28: Cross sections of a coil former with interleaved primary and two secon-daries. Second secondary winding using single wire on the left and using two parallelwires on the right.

Figure 28 presents a construction with margins where the required creepage distanceis produced by margin tape barriers in the sides of coil former. Different layers andthe outer half of the primary are insulated by tape. The construction on the rightin Figure 28 uses two parallel wires next to each other in the second secondary.Wire areas of all windings are equal in both constructions. As can be noted from

43

Figure 28, using parallel wires leads to a more compact design of the windings andthus, the leakage inductance of the transformer is reduced.

2.3.8 Manufacturing process

In order to design effectively manufacturable flyback transformers, the designershould be aware of the typical manufacturing process of the component. Follow-ing information is based on the description of a contract manufacturer of magneticcomponents.

Magnetic components are manufactured according to the specifications of a cus-tomer. The contract manufacturers may have preferences for the suppliers of differ-ent parts. Due to higher volumes, the parts from the preferred suppliers are usuallycheaper or their lead times are shorter than those from other suppliers.

Windings are typically constructed using coil formers. In the winding operation,the coil former is attached to the rotating shaft of a spooling machine. The machinerotates the coil former as the specified winding wire is guided to the coil former.The correct number of turns is ensured by entering the specified number of turnsto the spooling machine. Furthermore, the specified number of insulation tapelayers between winding layers are ensured by the corresponding number of rotations.Depending on the component, layers comprised of approximately six turns or less aredone as handwork. In addition, margin tapes, fixing the wires to pins of coil formersand other special constructions are made by hand. The winding construction beginsfrom the bare coil former and ends to the outermost winding and wrapper insulationlayer.

After the winding operation, the pins of coil formers are dipped into the solderpot for a very short period of time to prevent melting of the coil former. Afterthe soldering, the core halves are attached upon the spooled coil former and fixedtogether by glueing or by using mounting clips. If varnishing is specified, the compo-nent will be dipped to lacquer. Varnishing of magnetic components is usually done tofurther enhance the insulation, to enhance conduction of heat from windings to core,to reduce noise generated by the component or to protect the component from thepossible pollution of the environment. Finally, the specified electrical and insulationtests are conducted on the complete component to verify the specified functioning.

This chapter has presented the magnetic theory, the most important parametersand the notable issues to design flyback transformers. In the following chapter,flyback transformers are designed for two different flyback converters according tothe theory presented in the previous chapters.

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3 Design of flyback transformer

In this chapter, flyback transformers are designed for one DC-to-DC and one AC-to-DC flyback converter. The iterative design process is based on the theory presentedin the previous chapters.

The design process begins with determining the initial values and preferencesof the designs. This is followed by determining the parameters of the componentssuch as turns ratios, inductances, numbers of turns and allowed thermal rise. Then,materials and the structure of the designs are decided based on the determinedparameters and the preferences. The design process ends with analyzing the costsof the different designs. All calculations are presented in Appendix C excludingstraightforward parameters, which are determined directly in the text.

3.1 Initial values and design preferences

The initial values are determined from the specification of the auxiliary power supply.Moreover, the auxiliary power supply has two independent flyback converters. Thedouble ended topology presented in 2.2.5 is used in the DC-to-DC converter and thedesigns of this converter will be marked as DCDC from now on. The single switchtopology presented in 2.2.4 is used in the AC-to-DC converter and the designs of thisconverter will be marked as ACDC. The input and output voltages and the powerdemands of the two converters are listed in Table 3.

Table 3: Voltages and power demands of the converters.Converter VD (V) VO1 (V) VO2 (V) IO1 (A) IO2 (A) PO (W)

DCDC 240–1100 24 ± 5% 24 ± 10% 1.8 1.2 72ACDC 390–806 24 ± 5% 24 ± 10% 0.41 0.45 21

In Table 3 the input voltage VD of the AC-to-DC converter is assumed to be com-pletely smoothened by the input filter capacitor and the power demand of the loadPO is simply the sum of the both output powers. Fruthermore, the output voltageVO1 is regulated in both converters and has tighter tolerance than VO2. Moreover,the output VO2 is considered as user potential, which means that user has access tothis potential.

The used controller circuit is the QR type UCC28600 from Texas instruments.The operation of the circuit is described in 2.2.6 and the relating specificationsneeded in designing the flyback transformer are summarized in Table 4.

Table 4: Specifications of UCC28600 controller circuitry to design flyback trans-formers.

fsw (kHz) vCC (V) vFCC (V) vAUX (V)40–130 15.3–21 0.8 16.1–21.8

In Table 4, the vFCC is the forward voltage drop of the used diode and it is addedto vCC to obtain the voltage in the auxiliary winding vAUX .

45

The maximum voltage stress over the secondary diodes is 300 V for the DC-to-DC converter and 600 V for the AC-to-DC converter and the maximum DC currentsare correspondingly 15 A and 3 A. Furthermore, the forward voltge drop vF of bothdiodes is 1 V.

Specifications of the MOSFET, used in the converters, are presented in Table 5.

Table 5: Specifications of the used MOSFET. [31]vrating (V) irating (A) RDS,on (Ω) cDS (pF)

1500 V 2.5 6 100

In Table 5, the resistance during on time is RDS,on, the vrating is the maximum ratedvoltage stress and the irating is the maximum rated current of the MOSFET.

Some properties of the converters are fixed, including the pin orders of the fly-back transformers on the PCB (printed circuit board). This delimits the usage ofpossible cores, as suitable coil formers are not available for all shapes. Furthermore,the minimization of leakage inductance is desirable because of the already existingvoltage snubbers. For these reasons, ETD cores are chosen to be used in the designs.

The values of toughest operating point, minimum input voltage and the maxi-mum load current, are used to calculate the parameters. Moreover, the maximumambient temperature of the auxiliary power supply is 65 C in the final applicationand 100C will be used as the operating temperature when calculating the temper-ature dependent parameters.

The focus of the designs will be reliable operation in all operating points andcost effectivity. Clearances and creepages are determined according to the safetystandard as described in 2.1.4 and the thermal rise of the component is limitedbelow the maximum temperatures of used materials. Moreover, the cost effectivityis taken into account by using TIW only in windings requiring reinforced insulation,by selecting chinese suppliers for cores and avoiding complex structures to minimizethe time used for manufacturing. Choosing the suppliers for wires and coil formersis left for the contract manufacturer because they usually have preferences for thelocal least expensive suppliers.

3.2 Turns ratio and inductance

The turns ratioNPS of the ACDC design is chosen according to the maximum voltagestress of the MOSFET by Equation (17) and by using a value of vspike = 0.3 ∗ VD.A safety margin of 30 % of the VD has to be left to the maximum rated voltage ofthe MOSFET because the actual value of the voltage spike vspike is not known [9, p.130]. Contrary to ACDC designs, in the double ended DCDC design, the turns ratiowill be decided according to the maximum achievable duty cycle of a QR controlledflyback as described in 2.2.6. This is because the input voltage range is as wide as240-1100 V. Due to the wide range, the duty cycle Dmax is needed to be as high aspossible, to maximize the on time with the maximum input voltage, to make surethe MOSFET will turn on completely.

46

The maximum duty cycle of the MOSFET Dmax is 1−Doff , where Doff is 0.5in BCM. In the QR mode, the Dres is initially 0.04, which should be additionallysubtracted from the maximum duty cycle as described in 2.2.6. Therefore, Dmax

is chosen to be 0.44. The Dmax is chosen 0.01 lower than theoretical maximum toensure that the converter does not enter to the continuous region. The value of Dres

is achieved by assuming that tres is initially 500 ns [1].In addition, 45 kHz will be used as the initial value of switching frequency in the

calculations of DCDC designs. This frequency is chosen according to the existingconverter design, in which for example EMI filters are optimized on the ground ofthis operating point. Conversely, the switching frequency of the ACDC design willbe 100kHz.

The voltage drop due to the resistance of the primary RP and RDS,on is unknownin the beginning, because the rms value of the primary current is unknown. There-fore, the calculation of the inductance L will be iterated twice by first neglecting thevdrop and calculating the inductance and the peak current of primary winding. Then,the rms value of the current is calculated from the peak value using Equation (54)and an approximation of the voltage drop is attained by vdrop = (RDS,on+R)ip,rms [7,p. 807-808]. The calculated value approaches the actual vdrop as the number of itera-tions increase. In addition, the resistance of the primary winding R is approximatedto be 1 Ω in the calculation.

The initially calculated values are presented in Table 6.

Table 6: Calculated NPS, L, Dmax, iPpk, iP,rms and vdrop.Design NPS L (mH) Dmax iPpk (A) iP,rms (A) vdrop (V)DCDC 7.86 1.49 0.43 1.50 0.57 7ACDC 8.42 2.79 0.29 0.38 0.12 0.84

Because the approximated vdrop of the ACDC design, shown in Table 6, is only lessthan 1 V, it is neglected in the calculations.

3.3 Core size and flux density

The material and the size of the core are initially selected from Table 1. Table 1presents the suggested sizes and shapes of cores for the switching frequency of100 kHz. However, the parameters for DCDC design in Table 6 were calculatedwith 45 kHz as the operating frequency which should be noticed when selecting thesize of the core. The lower operating frequency means that higher energy is stored tothe air gap during one switching cycle. From Table 6, ETD34 is chosen for DC-to-DC converter and ETD29 is chosen for AC-to-DC converter according to the outputpower demands of the converters.

The stored energy per one switching cycle and the maximum stored energy ofthe selected cores are shown in Table 7. In Table 7, the maximum throughput of thecores Emax is determined from Figure 26 with maximum lengths of air gaps set to 10percent of dcp in Table 2. The energies E in Table 7 have been calculated in AppendixC for both converters and they appear to be less than the maximum energies Emax

47

of the cores. To be noted from Table 7, the ETD29 seems to be marginally suitablefor the DCDC design because the energy of DCDC design E = 3.36 mJ is less thanEmax = 3.4 mJ.

Table 7: Initially calculated energy per one switching cycle, the maximum energythroughput of the selected cores and the used maximum lengths of air gaps.

Design E (mJ) Emax (mJ) lg (mm)DCDC34 3.36 4.2 1.11ACDC29 0.53 3.4 0.98

As a result, an alternative size of the DCDC will be designed using the sizeETD29 and the different designs will be marked as DCDC29, DCDC34 and ACDC29according to the size of the selected core.

Equivalent materials to 3C30, Mn-Zn ferrite from Ferroxcube, were chosen tobe used in both designs because the material is intended for flyback transformersoperated with frequencies less than 200 kHz as described in 2.3.4. The Chinesemanufacturers DMEGC and TDG provide materials with properties close to 3C30and they are chosen as the two suppliers of the cores. The properties of the materials,typically needed in design process, are collected to Table 8 from Figure B1, Figure B2and Figure B3.

Table 8: Collected properties of materials 3C30, DMR40 and TP4.Material TC (C) µi Bsat (mT) PV (mW/cm3)

3C30 >220 2100 ±20% 440 440DMR40 >215 2300 ±20% 400 410

TP4 >220 2300 ±25% 390 410

In Table 8, the µi is the initial permeability, which is the permeability of the corewith a low field strength and the value is typically reported by manufacturers in thedatasheets.

If the properties of Table 8 are compared to each other, the similarities betweenthe different materials can be noticed. The value of Bsat in Table 8 is the reportedvalue in 100 C and the power loss density PV is the value with 200 mT peak acflux density and 100 kHz operating frequency. Moreover, the peak ac flux densityin DCM is half of the peak to peak flux density swing ∆B [18].

The value of the ∆B should be chosen less than Bsat to avoid saturating thecore with the minimum input voltage. For ferrites, such as the materials selected, atypically used value for ∆B is 0.25-0.30 T [18]. The desired value of the flux densityswing ∆B is initially selected to be 0.25 T. The lower operating flux density leadsto lower core losses as can be seen from the graph on the left in Figure B1.

3.4 Numbers of turns, air gap and peak current

In order to achieve the flux density swing of 0.25 T, the minimum number of turnswill be calculated with Equation (56). The correct length of air gap lg is chosen by

48

iterating Equation (51) with different air gap lengths lg until the calculated LM isclose to the desired value of inductance in Table 6.

The minimum number of turns for the DCDC34 is 91.6. Hence, the selectednumber of turns is 92. Moreover, the minimum calculated number of turns forDCDC29 is 116.2 but the chosen number of turns will be the same 92 to maintainsimple comparison between the different sizes. This results in an increased fluxdensity swing to approximately 0.32 T but because the chosen value of inductanceis less than that calculated initially, the peak value of flux swing will be closer to0.28 T. Furthermore, the increased flux density increases the core losses but is nothigh enough to saturate the core.

The number of turns in the secondaries of the DCDC34 and DCDC29 is de-termined according to Equation (57). If the chosen number of turns in primaryNP = 92 is divided by the turns ratio NPS = 7.86, the number of turns in thesecondaries is NS = NP

NPS≈ 11.7. Therefore, the turns in the secondaries is selected

to be 12, which results in the turns ratio NPS = 9212≈ 7.67.

The minimum allowed air gap length is chosen to be 0.3 mm due to practicaltolerances in the grinding of the gaps and the minimum calculated number of turnsfor the ACDC design is 57.1 turns. However, 57 turns with the 0.3 mm air gap lengthproduces only an inductance value of 1.1 mH with the selected core. Therefore, ahigher number of turns has to be chosen. 88 turns with the same air gap length,0.3 mm, produces the value of inductance 2.65 mH which is close enough to thedesired 2.79 mH.

The number of turns in the secondaries of the ACDC29 is determined equallyas the turns for DCDC designs. If the chosen number turns in primary NP = 88is divided by the turns ratio NPS = 8.42, the number of turns in the secondariesis NS = NP

NPS≈ 10.45. Therefore, the turns in the secondaries is selected to be 11,

which results in the turns ratio NPS = 8811

= 8. If the number of turns were chosento be 10 instead of 11, the turns ratio would have become larger than 8.42 and thereflected voltage would have grown from the selected value, narrowing the safetymargin to the maximum rated voltage of the MOSFET.

The number of turns in the auxiliary winding is calculated with Equation (21).The voltage of the auxiliary winding is designed to be approximately in the middleof the specified operating voltage range vAUX ≈ 19 V. The selected turns ratiosproduce lower voltage than 19 V to take into account the effect of cross regulation,which increases the voltages in the lightly loaded windings if compared to the moreheavily loaded regulated output.

The calculated and selected properties of the designs are presented in Table 9.

Table 9: Properties of the different designs calculated according to the desired values.Design NP NS N lg (mm) vAUX (V) LM (mH) ton (µs) iPpk (A)

DCDC34 92 12 9 1 18.8 1.39 8.88 1.49DCDC29 92 12 9 0.8 18.8 1.35 8.51 1.49ACDC29 88 11 8 0.3 18.2 2.65 2.84 0.42

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3.5 Allowed thermal rise and losses

The thermal rise of the designs is proportional to the area of the cooling surface asdescribed in 2.3.6. The size of the surface area can be calculated from the dimensionsof the cores shown in Figure 29. The different dimensions of the two cores, shownin Figure 29, are presented in Table 10.

W

Z

Y

Y

X

Figure 29: Dimensions of a flyback transformer.

Table 10: Dimensions of Figure 29. [19, p. 541,545]ETD core W (mm) Z (mm) X (mm) Y (mm)

34 35 34.6 11.1 5.529 30.6 31.6 9.8 5

The areas and allowed losses of both sizes are calculated assuming the maximumtemperature rise ∆T of 50C. This should result in the maximum temperature of115C if the ambient is 65C. Moreover, the window of the core is assumed to be fullwhen calculating the area of the surface. Equation (68) is used in the determinationof the allowed losses in Appendix C and the calculated areas and losses are presentedin Table 11.As can be noticed from Table 11, the allowed losses are higher for the larger designbecause of the larger cooling surface. On the other hand, if the losses are equal, thetemperature rise of the larger design is lower.

50

Table 11: The calculated areas and the allowed losses of the designs.ETD core A (cm2) Ploss (mW)

34 44.6 329829 35.0 2588

The allowed copper losses PCu can be calculated by subtracting core losses fromthe allowed total losses. The densities of the core losses are determined accordingto the used switching frequency from Figure B2 and Figure B3 in Appendix B andaccording to the peak value of the flux swing ∆B

2. In addition, the loss densities are

typically reported for certain temperatures and the used graph should be closest tothe operating temperature of the design. The chosen graph used in the designs isfor 100C and the values of ∆B are calculated in Appendix C.

Figure 30 shows the graph of Figure B3 in which the densities of core losses havebeen marked for different designs.

DCDC34

DCDC29

ACDC29

Flux density (T)

Co

re lo

ss d

en

sity

(kW

/m3)

1

10

100

1000

10.1

Figure 30: Densities of core losses in the designs using TP4 material from TDG [24].

The unit of the loss density in Figure 30 is kW/m3 which equals mW/cm3. Further-more, the loss density line, between the lines of 32 kHz and 64 kHz, is interpolated

51

to the graph to facilitate the determination of the values.The effective volume of the core Ve is calculated by multiplying the effective

length of the core lc by the cross sectional area of the core AC , which are checkedfrom Table 2. The value of core loss is calculated by multiplying the effective volumeof the core by the density of the core loss and the determined values of core andcopper losses of the designs are presented in Table 12.

Table 12: Densities of core losses, effective volumes of the cores and core and copperloss.

Design ρFe (mW/cm3) Ve (cm3) PFe (W) PCu (W)DCDC34 24 7.63 0.18 3.1DCDC29 43 5.53 0.24 2.3ACDC29 32 5.53 0.18 2.4

As can be seen from Table 12, the core losses are lower than 10% of the copperlosses in all designs. Therefore, if the temperature rise of the designs will be 50C,the copper losses is the major cause of the temperature rise.

3.6 Creepage margins and arrangement of windings

Clearance and creepage distances between different windings are needed accordingto the safety standard as described in 2.1.4. Reinforced insulation is required onlyin the second output where user can access. Hence, TIW is used in this windingto establish protective separation from other circuits. This means that creepagemargins are needed in the ends of coil former to provide functional insulation betweenthe other windings.

The impulse withstand voltage in Figure A2, for designing the clearances offunctional insulation, is defined according to the maximum voltage of the dc-to-dc converter and using the overvoltage category 2. The same interpolated impulsewithstand voltage, 4771 V, applies for both converters because they are consideredto belong into same circuit and this impulse withstand voltage defines more severerequirements for clearance than the temporary overvoltage of the mains circuit or thehighest working voltage. The required clearance distance, according to the 4771 Vimpulse withstand voltage, is approximately 4 mm.

The creepage distance is determined according to the maximum working voltagesof the converters. 1100 V is used for both converters as well as the pollution degreeof the printed wiring boards, which is 2. The interpolated creepage distance isapproximately 5.5 mm. As a result, 3 mm wide tape margins are fitted to bothends of coil formers. These margins ensure that the needed creepage distances isestablished between all windings. In addition, the required creepages and clearanceshave to be taken into account when selecting the coil formers and numbers of pins.

The structure of the windings has the effects to the resistance and the leakageinductance of the windings described in 2.3.7 and 2.3.3. In addition, the windingsshould be able to be spooled by machines to ensure the effortless construction of thewindings.

52

The primary is therefore divided into four layers each consisting of 23 turns perlayer. After the first layer is spooled, a single layer of insulation tape is wrappedon top of it to provide insulation. Then, the second layer of the primary is spooledbackwards to the same side where the first layer began. The secondaries are locatedbetween the two double layers of the primary.

The structure of the DCDC designs is presented in Figure 31. The ACDC designuses the same construction but has different numbers of turns.

Figure 31: Structure of the DCDC34 and the DCDC29.

The number of insulation tape layers is different between the layers in Figure 31.Two layers of tape is used between the primary and the first secondary, betweenthe auxiliary winding and the primary and on the top of the outermost layer ofthe primary to provide additional insulation. In addition, the auxiliary winding isspread to the entire width of other windings to increase coupling to the secondaries.Teflon tubing is used in the ends of wires to provide insulation when routing theends of the windings to the pins of the coil former.

The tape margins add complexity to the structure, but it is the price paid fordeciding not to use TIW in all windings. In addition, the margins narrow theavailable width of the coil former, which may increase the overall thickness of thewindings or the ohmic resistance, if the diameters of wires have to be decreased.

3.7 Diameters of wires and ohmic losses

In order to determine the diameters of winding wires, the rms values of currents haveto be calculated. The peak values of primary currents have already been presentedin Table 9 and the rms values are calculated using Equation (54). Moreover, thepeak values of the currents in the secondaries are calculated with Equation (16).

The calculated rms value of the current in the secondaries is further dividedbetween the two separate outputs, according to the relations of the DC values ofmaximum load currents. The calculated rms values of different designs are shownin Table 13.In Table 13, the currents iS1,rms and iS2,rms are the rms values of currents in thesecondaries and the itot is the sum of the rms currents with respect to the primary.

53

Table 13: Calculated rms values of the currents in the windings.Design iP,rms (A) iS,rms (A) iS1,rms (A) iS2,rms (A) itot (A)

DCDC34 0.57 4.7 2.8 1.9 1.2DCDC29 0.57 4.6 2.8 1.8 1.2ACDC29 0.13 1.44 0.69 0.75 0.31

The factor αi, calculated by Equation (58), depicts the percentual fraction of thewindow area allocated for the winding carrying current ii,rms. The cross sectionalareas of each winding wire is then determined according to Equation (59). Moreover,the fillling factor of the windowKu is approximated to be 0.33 for the wires consistingonly from one strand. This is higher than the typical values for Ku described in 2.3.4because the area occupied by the coil formers is already subtracted from the windowareas of the cores in Table 2, which are used in the calculations.

The calculated cross sectional areas and maximum diameters of winding wiresare presented in Table 14.

Table 14: Cross sectional areas and maximum diameters of wires.Design AwP

(mm2)AwS1

(mm2)AwS2

(mm2)dwP(mm)

dwS1

(mm)dwS2

(mm)DCDC34 0.22 1 0.68 0.53 1.13 0.93DCDC29 0.15 0.74 0.49 0.44 0.97 0.79ACDC29 0.14 0.76 0.81 0.42 0.98 1.0

Two different wire types are chosen for each design. The first type is a wiremade of a single conductor and the second type is a wire consisting of multiplestrands to reduce the AC resistance due to high frequency components in the current.Moreover, the diameter of the strands of these Litz type wires is chosen according tothe skin depth at 160 kHz, which is calculated using Equation (67). Furthermore,the skin depth at this frequency is selected due to the power losses at the secondharmonic of the switching frequency in the intended most common operating pointof the dc-to-dc converter [26]. The designs using Litz wire will be marked by addingsubscript m to the name of the design.

The diameters of wires are chosen according to the maximum values in Table 14and by calculating the equivalent number of strands with Equation (62). Neverthe-less, the filling factor of Litz type wires is lower than that of a single conductor. Thisis taken into account by selecting smaller number of strands than calculated. Theselected diameters of wires and numbers of strands are presented in Table 15. Thefinal numbers and the diameters of strands in Table 15 were chosen after iteratingdifferent diameters by drawing the window area using correct dimensions of coil for-mers, margin barriers, insulation layers, wire insulation and diameters of wires. Thedrawing of the DCDC34m, with the chosen winding wires, is presented in Figure 31.In addition, the availability of Litz type TIW for the second secondaries delimitedthe maximum diameter of one strand to 0.3 mm. For this reason, the maximumdiameter of strands of all Litz type wires was chosen to be 0.3 mm.

54

Table 15: Chosen diameters of conductors and numbers of strands.Design dwP,p

(mm)pP dwS1,p

(mm)pS1 dwS2,p

(mm)pS2

DCDC34 0.55 1 1 1 0.9 1DCDC34m 0.25 3 0.3 8 0.3 7DCDC29 0.4 1 0.9 1 0.7 1

DCDC29m 0.2 3 0.28 7 0.23 7ACDC29 0.4 1 0.8 1 0.9 1

ACDC29m 0.2 3 0.28 6 0.27 7

The calculated skin depth at 160 kHz is 0.19 mm and the maximum diameterbased on this value would be 0.38 mm. As a result, the selected maximum diameterof strands, in the Litz type wires, corresponds to the skin depth at 258 kHz.

The ohmic losses of windings are calculated according to Equation (64) and theohmic resistance is determined by Equation (65). In order to determine the resis-tances, the mean lengths per turn have to be calculated. Furthermore, to calculatethe mean lengths per one turn, the radius of the coils are determined according to thedistances of different windings in Figure C1. The calculated ohmic resistances andcurrent densities of windings and the ohmic losses of different designs are presentedin Table 16.

Table 16: Calculated ohmic resistances of the windings, current densities in thewindings and the ohmic copper losses of the designs.

Design R0P

(Ω)R0S1

(Ω)R0S1

(Ω)JP(A/cm2)

JS1

(A/cm2)JS2

(A/cm2)PCu0

(W)DCDC34 0.53 0.019 0.027 240 357 299 0.42

DCDC34m 0.86 0.027 0.035 387 495 384 0.62DCDC29 0.89 0.021 0.040 454 440 468 0.58

DCDC29m 1.19 0.031 0.052 605 650 619 0.80ACDC29 0.85 0.024 0.022 103 137 118 0.04

ACDC29m 1.13 0.033 0.035 138 187 187 0.05

As can be seen from Table 16 the values of ohmic copper losses are a minor fractionfrom the allowed total copper losses of Table 12. This means that AC losses isthe major part of the copper losses or on the other hand, the temperature of thecomponent will not rise 50C. Moreover, the current densities and the calculatedohmic losses of both ACDC designs are low compared to the DCDC designs and thecurrent density in the DCDC29m design exceeds the typical maximum of 500 A/cm2

according to 2.3.5.

3.8 Costs

The designed flyback transformers were ordered from a contract manufacturer. Thecomponents were manufactured according to the specifications which are based on

55

the calculated and selected properties of the designs. The relative costs of the designsare shown in Table 17.

Table 17: Costs of complete designs. The cost of the most expensive design equals100 %.

DCDC34(%)

DCDC34m

(%)DCDC29(%)

DCDC29m

(%)ACDC29(%)

ACDC29m

(%)85 100 62 77 69 79

The largest designs are the most expensive, as can be seen from Table 17. This isdue to the higher amount of used materials if compared to the smaller designs. Inaddition, designs in which Litz wire is used, are more expensive than the otherwisesimilar designs in which the windings are made from single conductors.

In order to find out what are the most expensive materials in the designs, thecosts are broken down to show the portions of the different materials. The relationsof different costs to the costs of the complete components are presented in Table 18.

Table 18: Costs of different materials in relation to the cost of the complete compo-nent.

Design Core(%)

Coilformer(%)

Wire P(%)

Wire S1(%)

TIWS2(%)

Labor(%)

Othercosts(%)

DCDC34 22.5 9.7 10.8 4.7 13.2 29.5 9.6DCDC34m 19.0 8.2 8.7 3.6 32.4 19.9 8.2DCDC29 15.6 10.7 6.7 4.5 10.3 40.1 12.1

DCDC29m 12.5 8.6 4.8 3.4 28.7 32.2 9.8ACDC29 14.0 9.7 6.0 3.0 13.0 43.3 11

ACDC29m 12.2 8.4 4.6 2.4 33.7 29.3 9.4

In Table 18, the other costs include the costs of insulation tape, margin tape, epoxy,wire of auxiliary winding, teflon tubing and tin used for soldering. Furthermore,the standardized diameters of enamelled wire, shown in Table 15, are used in theprimary and in the secondary S1.

Table 18 shows that the proportion of the Litz type TIW, used in the secondsecondaries, is approximately 30 % of the complete component, although it was onlyused in one secondary winding of 11 or 12 turns depending on the design. Moreover,the single conductor TIW seems to be approximately 60 to 70 % cheaper than theLitz type TIW. On the other hand, the usage of Litz type wires has an effect on themanufacturing time of the component if the diameter of the used wire is close to1 mm or over. This is because single conductors of such diameter are beginning tobe stiff, which affects the time used for manufacturing the component and therefore,the costs of labor. If all windings would have been completely made of TIW, thecosts of labor and other costs would have decreased but the costs of windings wouldhave increased more. Contrary to TIW, the cost of Litz type enamelled wire wasnot higher than the cost of the enamelled wire consisting of the single conductor.

56

In addition to TIW, an important part affecting the overall cost of the componentis the size of the core. The ETD34 core, made of TP4 material by TDG, was twiceas expensive as the ETD29 used in the smaller designs.

For a comparison to costs in Table 18, the costs of DCDC34m is calculated usingLitz type TIW in all windings. The cost of the primary is calculated using the TIWof DCDC29m design and scaling the cost with the relation of the mean lengths perturns to the DCDC34m and finally multiplying by the turns ratio 7.67. Furthermore,the same TIW, as in the second secondary of DCDC34m, is used in both secondariesbut the costs are scaled by the relations of the mean length per turns. In addition,the labor costs are assumed to decrease by a third and other costs by 10% becausethe margin tapes are no longer needed for creepages in the construction.

The calculations are shown in Appendix C and the costs of the materials andthe complete design are presented in Table 19.

Table 19: Costs of the different materials and the complete design, in relation tothe cost of the DCDC34m design in Table 17, if TIW is used in all windings ofDCDC34m.

Core(%)

Coilformer(%)

Wire P(%)

Wire S(%)

Labor(%)

Othercosts(%)

Complete(%)

19.0 8.2 183 60.7 13.3 7.4 292

As can be seen from Table 19, the costs of windings increase significantly and thecost of the complete component increases to 292 % of the cost in Table 17. On theother hand, the calculation does not take into account the possible reductions in theprices of TIW which may result from the increased amount used in the components.

The advantage of using TIW is that a smaller size of the core could possibly bechosen because the creepage margins are no longer needed. However, the thicknessof the windings can become a problem with multiple secondaries.

The design process of the flyback transformers was presented in this chapter.The process began by determining the initial values and was followed by calculatingthe properties and selecting the materials of the components. The process endedto breaking down the costs of the materials. The suitability of the designs for theintended applications will be verified in the next chapter.

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4 Verifications of flyback transformers

In this chapter, the operation of the designed flyback transformers is verified by mea-surements. The magnetizing and leakage inductances of primary and resistances ofall windings are measured as function of frequency. Furthermore, the currents satu-rating the cores of the designs and maximum voltage stress of the secondary diodesare found out. In addition, the operation of the designs is tested in the intendedapplication with typical minimum and maximum loads. Finally, the thermal rise ofthe designs will be verified with the maximum load currents.

The methods and the equipment of the different tests are introduced beforepresenting the results. The results are compared to the theory and calculated values,presented in the previous chapters.

4.1 Measurement of inductances

The magnetizing and leakage inductances between primary and secondaries weredetermined with Agilent 4294A precision impedance analyser, which measures thecomplex impedance of the DUT (device under test) as function of the selected fre-quency range. Figure 32 shows the measured impedance, from the primary, of asample of DCDC29 design.

Resonance peak

Inductive operation

Capacitive operation

Figure 32: Impedance curve of a DCDC29 design.

Three different characteristic regions can be noted from the magnitude curve ofFigure 32. The first one is the region of inductive operation. This region is thelinearly increasing part of the curve at frequencies lower than the resonance peak.The second one is the resonance in which the inductance and the capacitance of the

58

inductor are in series resonance. The third is the region of capacitive operation wherethe magnitude of the impedance decreases linearly. In addition to the magnitudecurve, the phase curve shows the shift of the operation from inductive to capacitiveat the frequency of the resonance. [32]

The value of inductances are calculated from the measured impedance accordingto

L = sin(θ)|Zx|2πf

(69)

where θ is the phase angle of the impedance in degrees, |Zx| is the magnitude of theimpedance and f is the frequency. The frequency range of the calculated inductanceis selected to be in the linearly increasing region of the magnitude curve.

The 4294A precision impedance analyzer uses auto-balancing bridge method tomeasure impedance. In the method, the DUT is connected between the high andlow terminals and the potential of the low terminal is maintained at zero volts. Thepotential of the high terminal is measured directly by a vector volt meter while thecurrent flowing through the DUT is determined by measuring the voltage with avector volt meter over a range resistor Rr. Consecuently, the complex impedance iscalculated by

Zx =vxvx

= RrvxvR

(70)

where vx is the voltage over the DUT, ix is the current through the DUT, Rr is thevalue of the range resistor and the vR is the voltage over the range resistor. [32]

The value of the range resistor determines the measuring range and therefore,the accuracy of the measurement. The value of the resistor is known and it ischanged automatically based on the impedance of the DUT. Figure 33 presents theconfiguration of the measurement method.

Figure 33: Impedance measurement using auto-balancing bridge method. [32]

In Figure 33, the oscillator generates a signal leading to the currents ix through theDUT and ir through the range resistor Rr. Because the potential of the low terminalis zero volts, the two currents are equal. [32]

59

The inductance of the components is measured by connecting one end of the pri-mary to the high and the other end to the low terminal of Agilent 16047E test fixture,which is attached to the precision impedance analyzer. Coupling from primary tothe test fixture was ensured by soldering approximately 7 cm long copper strips tothe pins of primaries and attaching the strips to the high and low terminals by thescrew clamps of the fixture. Figure 34 presents the coupling of the components tothe test fixture.

Figure 34: DCDC34 attached to the test fixture by copper strips.

When measuring the magnetizing inductance, all secondaries, including auxiliarywindings, were left open. Contrary to magnetizing inductance, leakage inductanceswere measured from the primary by short circuiting all the secondaries. [33]

The precision impedance analyzer was turned on approximately 30 minutes be-fore conducting the measurements in a typical room temperature. In addition, theanalyzer was always calibrated before conducting the measurements by measuringthe impedance of open circuit and then by short circuiting the high and low ter-minals with a gold plated piece of metal. The measured impedance of the piece isshown in Figure 35. Figure 35 shows that noise limits the accuracy of the analyzerto approximately 1 mΩ.

Curves presenting typical magnetizing inductances are shown in Figure 36. Theinductances of DCDC29 in the curves of Figure 36 are calculated with Equation (71)from the measured impedances. Furthermore, the accuracy of the precision impedanceanalyzer is at least ±0.1 % of the measured values in a frequency range from 10 kHzto 300 kHz and with a 0.5 V rms signal of the oscillator [34].

The curves in Figure 36 are flat from the 10 kHz to approximately 100 kHzuntil the inductance begins to increase. The increase is a consequence of the seriesresonance occuring at a higher frequency.

The curves of other designs are not presented because the trends are close to that

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Figure 35: Magnitude of the measured impedance of the short circuiting piece.

seen in Figure 36 and the increase of the inductances, due to the resonance, do notlead to problems in the application. Instead, the mean values of mutual inductances,calculated for 5 samples at 100 kHz, scattering of the values and the differences tothe calculated values of 3.4 are presented in Table 20.

Figure 36: The magnetizing inductances of DCDC29 calculated from the measuredimpedances of 5 samples.

In Table 20, the highest difference to the calculated mutual inductances of 3.4 is6 %, which means that the measured values are generally well in line with the cal-culated values. The differences may originate from differences in the actual lengthsof air gaps or differences between calculated and actual complete numbers of turns.In addition, the maximum scattering between different samples is 2 %, which meansthat structural differences between the samples are small. Furthermore, if the corescome from the same production run, the differences in the dimensions and the prop-erties of materials of the cores should be small between the samples.

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Table 20: Measured mutual inductances at 100 kHz, scattering between differentsamples and differences to the calculated values of LM of the previous chapter.

Design LM (mH) Scattering (%) Difference (%)DCDC34 1.40 ±1 +1

DCDC34m 1.43 ±1 +3DCDC29 1.33 ±1 -1

DCDC29m 1.27 ±2 -6ACDC29 2.73 ±1 +3

ACDC29m 2.58 ±1 -3

The leakage inductances of the primaries are measured only from a single sampleto find out differences between designs. Moreover, the accuracy of the precisionimpedance analyzer is at least ±1 % of the measured values in the frequency rangefrom 10 kHz to 1 MHz and with a 0.5 V rms signal of the oscillator [34].

The DCDC designs are compared to a flyback transformer, referred to DCDC39,which has the same number of turns in the primary and in the secondaries as inthe designed components. Furthermore, the DCDC39 is constructed using ETD39core size and the interleaved windings are all made of Litz type TIW. The measuredleakage inductances of the DCDC designs are presented in Figure 37.

Figure 37: Measured leakage inductances of the DCDC designs.

The measured leakage inductances of all designs, excluding DCDC39, are be-low 2 % of the measured mutual inductances. In addition, the trend of the curvesin Figure 37 are decreasing as the frequency increases, which is a consequence ofproximity effect between the windings [33]. Furthermore, this is more pronouncedin the DCDC34 and DCDC29 in which the windings are made of single conductorsof larger diameter than in the other designs. Moreover, the leakage inductance issomewhat greater in the DCDC39 design, than in the others, because the distancebetween secondaries and the primary is larger due to the additional thickness of the

62

wire insulations. This leads to a worse coupling between the primary and the sec-ondaries which can be noted as the higher leakage inductance in the measurements,as described in 2.3.3, if compared to other designs.

The leakage inductances of the ACDC designs are compared to a design which isconstructed using ETD34 core size and in which the primary is made of six layers ofLitz type enamelled wire. Furthermore, the windings are interleaved and the designis referred to ACDC34. The measured leakage inductances of the ACDC designs arepresented in Figure 38.

Figure 38: Measured leakage inductances of the ACDC designs.

In Figure 38, the leakage inductances of both ACDC29 designs are below 1 %of the measured mutual inductances. In addition, as with the DCDC designs, theproximity effect can be noted from results of ACDC designs. Furthermore, themeasured leakage inductance of the ACDC34 is higher than that of other two designs.This is a consequence of the two additional layers in the primary which makes theoverall thickness of the primary greater than in the two other designs. Moreover,the effect of the series resonance to the leakage inductances can not be seen fromFigure 37 or from Figure 38 because the proximity effect covers the increase of theleakage inductance in the observed frequency range and the series resonances occurat higher freuquencies than with the mutual inductance.

4.2 Measurement of resistances

The resistances of the designs are measured with the same equipment as the in-ductances. Moreover, the resistances are calculated from the measured impedancesby

R = cos(θ)|Zx| (71)

where |Zx| is the magnitude of the measured impedance.

63

Resistances of windings were measured only from a single sample because thestructural differences were assumed to be small and their effect to the resistancesis not relevant. The measurements were conducted on one winding at a time withother windings left open. Furthermore, approximately 7 cm long copper strips weresoldered to the pins of the windings to ensure a good coupling between the com-ponent and the test fixture. Furthermore, the measurements were conducted in atypical room temperature.

Figure 39 shows the typical behaviour of the measured resistances as function offrequency.

Figure 39: Measured trends of primary resistances of DCDC34 and DCDC34m

.

The increase of the resistance can be noted from Figure 39 as the frequency increases.This results mainly from skin effect, which was described in 2.3.6. Moreover, theeffect is less pronounced in the winding of DCDC34m because the windings of thedesign are made of Litz wire, which reduces the skin effect and the total AC resis-tance.

The trends of the resistance curves of primaries and secondaries are similar tothe curves seen in Figure 39. Therefore, only the measured resistances and theaccuracies, at 45 kHz, are presented in Table 21 and at 100 kHz in Table 22.

The accuracies in Table 21 and Table 22 are the accuracies of the precisionimpedance analyzer with a 0.5 V rms signal of the oscillator [34]. The measuredresistances are generally higher than the calculated ohmic resistances in Table 16.This results from the effects of AC resistance, which are already visible in the valuesof Table 21. Moreover, the resistances of the windings, made of Litz wire, areapproximately equally lower in Table 21 and in Table 22 than the resistances of thewindings made of single conductors. In addition, the resistances in Table 21 arelower than the resistances in Table 22, which is a consequence of the higher effectsof AC resistance with the increased frequency.

The higher resistance as the frequency increases refers to a higher thermal rise ofthe component due to the increased copper losses. Because the operating frequency

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Table 21: Measured resistances of primaries and secondaries at 45 kHz.Design RP

(Ω)Accuracy(Ω)

RS1

(Ω)Accuracy(Ω)

RS2

(Ω)Accuracy(Ω)

DCDC34 7.11 ±0.32 0.136 ±0.019 0.193 ±0.023DCDC34m 1.71 ±0.32 0.049 ±0.023 0.064 ±0.023DCDC29 4.57 ±0.30 0.089 ±0.019 0.145 ±0.020

DCDC29m 1.52 ±0.29 0.054 ±0.019 0.067 ±0.019ACDC29 3.66 ±0.62 0.080 ±0.012 0.108 ±0.012

ACDC29m 1.85 ±0.58 0.056 ±0.011 0.061 ±0.012

Table 22: Measured resistances of primaries and secondaries at 100 kHz.Design RP

(Ω)Accuracy(Ω)

RS1

(Ω)Accuracy(Ω)

RS2

(Ω)Accuracy(Ω)

DCDC34 20.07 ±0.71 0.371 ±0.015 0.463 ±0.015DCDC34m 5.85 ±0.72 0.123 ±0.015 0.168 ±0.016DCDC29 12.99 ±0.67 0.234 ±0.014 0.331 ±0.014

DCDC29m 3.77 ±0.66 0.098 ±0.014 0.131 ±0.014ACDC29 10.99 ±1.40 0.199 ±0.028 0.272 ±0.028

ACDC29m 4.83 ±1.32 0.113 ±0.026 0.134 ±0.026

of the flyback transformers will increase as function of input voltage, as describedin 2.2.6, the thermal rise of the component will be higher as the input voltage isincreased.

The AC resistance of the windings, during the actual operation in the intendedapplication, will be higher than what was measured. This results from the higher fluxdensity in the core, which will increase the fringing flux and therefore the resistancein the windings close to the air gap. In addition, the proximity effect will increasethe resistance as the conductors of different layers form higher surrounding magneticfield strengths around conductors than during the measurements.

4.3 Operation in different operating points

The operation of the components is verified by first measuring the saturation be-haviour of the designs and then as part of the intended flyback converters, mountedon the PCB. Moreover, the tests in the intended application include verifying theoperation with the lowest and highest specified input voltages of the converters. Fur-thermore, the operation of the components is verified with minimum and maximumload currents with the both input voltages. In addition, the voltage stress over thefreewheeling diode, in the secondary S1, will be measured to verify that the ratedvoltage of the diodes is not exceeded.

The current saturating the components is measured with a power choke testerDPG-10-1500A from ed-k. Furthermore, the saturation behaviour was only mea-sured from one sample of all designs. During the measurements, the DUT was

65

connected to the power choke tester with one meter long test leads. The connectionto the copper strips, soldered to the pins of primaries, was made using alligator clips.This type of connection might add inductance to the measurements and because ofthat, these type of test leads are not recommended by the manufacturer for mea-suring inductances of less than 10 µH [35]. Moreover, the other windings were leftopen during the measurements.

DPG-10 measures voltage over the DUT and current flowing through the DUT.The device calculates the inductance of the DUT as function of current from theslew rate di

dtof the measured current, according to the applied volt seconds. Before

the measurements, the voltage of the measurement pulse is selected from the typicaloperational range of the converter and the maximum duration of the pulse is set highenough to let the rising current saturate the core. furthermore, the measurementwas conducted on the components in a typical room temperature. The measuredsaturation curves of the components are presented in Figure 40.

Figure 40: Measured saturation behaviour of the designs. The results of DCDCdesigns above and ACDC designs below.

In Figure 40, the cores begin to enter saturation with a value of the current atthe beginning of the linearly decreasing part of the inductance curves. These points

66

are marked to the curves by black lines. Moreover, the accuracy of the current inFigure 40 is less than ±3 % of the values [35].

Because the measurement was conducted in a room temperature, the actualsaturation behaviour may be different and the cores may enter saturation with alower current than in the room temperature. This can be assumed by comparingthe amplitude permeability curves of the materials in the room temperature andin the temperature of 100C. For example, DMR40 material in Figure B2, whichhas similar characteristics than the TP4 material, will enter saturation with a lowercurrent in the higher temperature.

The value of the voltage pulse, used in the measurements, was 240 V for DCDCdesigns and 390 V for ACDC designs. According to the measured curves in Figure 40,the current saturating the cores is higher than the calculated peak values of primarycurrents in 3.4. The peak values of calculated primary currents are 1.49 A for DCDCdesigns and 0.42 A for ACDC designs. This means that all the components shouldoperate without problems at least in the room temperature.

The differences between the similar designs may result from differences in thegrinding of the air gaps or in other dimensions of the cores. In addition, the actualnumber of complete turns has an effect on the saturation behaviour. Furthermore,the ripple in the inductance occurs from the quality of the generated pulse of thepower choke tester with the chosen voltage.

The operation with minimum and maximum input voltages was tested on a PCBin a laboratory environment. The PCB was not mounted to the PV inverter due topractical reasons and therefore the verification was not done as part of the final actualapplication. Moreover, the temperature in the laboratory was the typical roomtemperature during the measurements. The vertical resolution of the oscilloscope is8 bits which determines the accuracy of the results obtained from the screen capturesof the oscilloscope [36]. The equipment, used in the measurements, are presented inTable 23.

Table 23: Equipment used for verifying the operation on PCB.Manufacturer Type Description

Tektronix DPO4034D OscilloscopeFug MCA 750 - 1500 DC power supplyEA EL 3400-25 Electronic loadTTi LD300 Electronic load

Trafox 3kLS-9.6k-400-0-400 Adjustable transformerTestec 2.5kV x100 Passive voltage probe

Tektronix N2890A Passive voltage probe

The DC power supply provides the controllable input voltage and feeds the cur-rent demand for the DC-to-DC converter. Furthermore, the controllable electronicloads are connected to the outputs of the converters and the passive voltage probesare used to measure the vDS of the MOSFET and value of the currents in the pri-maries. Moreover, the current is measured by measuring a voltage over a shuntresistor of known value.

67

Figure 41 presents the measured vDS and iP of the DCDC29m operating withthe minimum input voltage of 240 V and maximum output current of 3 A.

Figure 41: Measured waveforms of the DCDC29m with minimum input voltage andmaximum output current of 3 A. Yellow denotes vDS and blue denotes iP .

Typical characteristics of the double ended flyback converter circuit, described in2.2.6, can be noted from the measured waveforms of Figure 41. The converteroperates in the QR mode ensuring the highest duty ratio and switching in the firstwalley of the resonance that occurs between the primary inductance and parasiticcDS capacitance of the MOSFET. The measured switching frequency of the converteris approximately 44 kHz, which is somewhat lower than the calculated 50 kHz inAppendix C. This might result from the fact that the operating modes were notaccurately preprogrammed for these designs, as they should have been, accordingto the datasheet of the control circuit [15]. In addition, the peak value of current inFigure 41, 1.67 A ±1 %, is higher than the calculated 1.49 A. The measured valueof peak current is limited to the measured value by the controller circuit, meaningthat the current can’t increase any higher, which means that the output power isdecreased. This was observed as decreased output voltages in the volt meters of theelectronic loads because the loads were set to current control mode.

The operation of the same design with maximum input voltage and 0.4 A loadcurrent is shown in Figure 42.

68

Figure 42: Measured waveforms of the DCDC29m with the 1100 V input voltageand 0.4 A output current. Yellow denotes vDS and blue denotes iP .

The converter operates in the green mode with the waveforms of voltage and currentshown in Figure 42. Moreover, the converter utilizes walley switching but the tres ishigher and duty ratio is smaller than in Figure 41. In the green mode, the convertergenerates bursts of switching periods with the minimum switching frequency asdescribed in 2.2.6.

Contrary to the DC-to-DC converter, the measured vDS of the AC-to-DC con-verter was higher than the input voltage as described in 2.2.4. Moreover, the max-imum measured vDS of ACDC designs was 1130 V ±1.7 % including the voltagespike with maximum input voltage of 750 V. The minimum voltage of the ac-to-dcconverter was produced by connecting the input of the converter to the adjustabletransformer that was connected to a three phase grid. The output voltage of thetransformer was then adjusted to the minimum value of the converter. However,the maximum voltage for the same converter was provided by connecting the inputto the DC power supply, because the maximum output voltage of the adjustabletransformer was only 400 V.

Because the observed operation of all designs is comparable to that presented inFigure 41 and Figure 42, the results of the designs are presented in Table 24.The accuracy of the measured peak currents is ±2 % of the value in Table 24. The

69

Table 24: The measured iPpk and fsw with minimum and maximum input voltagesand maximum load currents.

Design VDmin (V) iPpk (A) fsw (kHz) VDmax (V) iPpk (A) fsw (kHz)DCDC34 240 1.67 42 1100 1.09 85

DCDC34m 240 1.67 42 1100 1.07 84DCDC29 240 1.67 42 1100 1.10 88

DCDC29m 240 1.67 44 1100 1.10 91ACDC29 410 0.44 94 750 0.35 100

ACDC29m 410 0.44 98 750 0.40 102

operation mode of both converters was DCM with the maximum input voltages andwith the minimum voltage, the converter operated in the QR mode as described in2.2.6. In addition, the primary current of the dc-to-dc converters was limited by thecontrol circuit which resulted in lower than 24 V output voltage with the minimuminput voltage. However, increasing the input voltage to approximately 250 V, thepeak value of the current was no longer limited by the controller and the outputvoltages appeared as the 24 V. This does not affect the operation of the PV inverterin the final application beacuse the initial specified input voltage range is slightlywider than the actual operational input voltage range in the final application.

All designs operated without problems in all verified operating points. Neverthe-less, the measured switching frequencies of all DCDC designs in Table 24 are closerto the fmin of the controller than what was expected according to the calculations inAppendix C. This should be taken into account by increasing the length of air gap ordecreasing the number of turns, if the designs are further developed in future. Thisdecreases the primary inductance of the components and as a result, the convertersshould operate with a higher switching frequency. Moreover, the scattering betweenthe samples will probably increase as the number of manufactured components in-crease because of the different production runs of the cores. As a result, a largerfrequency margin should be designed to be sure that the operation at the minimumfrequency clamp is avoided.

The voltage stress over the diode in the secondary S1 was measured separatelywith the passive probe from Agilent. The measured voltage stress with the maximuminput voltages was more than 100 V less than the rated maximum voltage of thediodes with all designs. This safety margin is sufficient for the diodes used in theconverters.

4.4 Thermal rise with minimum and maximum input volt-ages

The thermal rise of the designs was measured with an infrared thermal camera FlirS60 ThermaCAM. The image of the camera was observed in real time and the finalimage, after the temperatures were stabilized, was saved for determination of thefinal temperatures. Moreover, the accuracy of the thermal camera is ±2 C and onecelsius should be added to that value to obtain the accuracy of the measurements

70

because of the variations in the ambient temperature [37]. In addition, emissivitiesof different surfaces add uncertainty to the measurements. Other equipment, usedin the measurement setup, are the presented in Table 23.

The measurement was conducted to the components in the ambient temperatureof 23 C. As a consequence, the results are not directly comparable to the finalapplication because the ambient temperature inside the PV inverter is 65 C.

The thermal rise of the component is measured using the same setup as for veri-fying the operation in different operating points without the forced convection of thefinal application. This results in a higher thermal rise, than in the final appliction,which should be taken into account when evaluating the results. Moreover, only thesurface temperatures of the components can be observed with the thermal camera.The hot spot temperature of the windings is in the inner layers of the winding nearthe air gap.

The measurement was conducted to the components with the specified maximumload currents and with the same minimum and maximum voltages used for verifyingthe operation in different operating points in 4.3. After the temperature of thecomponent had stabilized, additional 15 minutes were waited until the final imageswere saved to guarantee that the tempereatures did not increase any more. Theimages of the thermal camera, presenting the final temperatures of DCDC34m, isshown in Figure 43.

Figure 43: Final temperatures of DCDC34m. Minimum input voltage 240 V on leftand 1100 V on right.

The temperatures in the upper right cornes of the images in Figure 43 showthe final temperatures of the core and the winding. Moreover, the point presentingwhere the value is measured is denoted over the component in the images. Thenumber one denotes the temperature of the core and two denotes the winding. Inaddition, the point denoting the temperature of the winding was always placed onthe hottest spot of the winding in all measurements.

The final temperatures of all designs are presented in Table 25.Table 25 shows that the final temperatures of all designs are higher with the max-imum input voltage. This results from the measured increase of the AC resistance,

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Table 25: The final temperatures of the components with minimum and maximuminput voltages.

Design VDmin (V) TFe (C) TCu (C) VDmax (V) TFe (C) TCu (C)DCDC34 240 67 85 1100 85 113

DCDC34m 240 56 64 1100 66 80DCDC29 240 67 87 1100 83 111

DCDC29m 240 68 79 1100 78 91ACDC29 410 51 55 750 58 63

ACDC29m 410 50 52 750 56 57

with the higher frequency, as the input voltage is increased. In addition, the lossesin the cores increase if the operating frequency is increased, which can be noted fromthe datasheet of the ferrite material in Figure B3. Moreover, the thermal rise of thewindings is higher than the thermal rise of the core in all designs. This results fromthe lower portion of the core losses if compared to the copper losses.

The effect of the wire type used in windings is clearly notable from the results.In addition, although larger component, the thermal rise of DCDC34 is equal tothe smaller DCDC29, which is a result of the higher losses in DCDC34 due to thehigher AC resistance. Moreover, the thermal rise of the windings, made of Litz wire,is lower than that of the windings made of single conductors. The difference is notas large between the ACDC designs as it is between the DCDC designs and the bothACDC designs are usable in the final application according to these measurements.

The absolute maximum temperature of the designs is the rated temperature,130C, of the insulation tape and insulation of the used TIW. In addition, somemarginal should be left to the rated temperature. The thermal rise of the DCDC34m

is the least high of the DCDC designs. Although, if approximately 40C is addedto the saturated temperatures, the temperatures of the other DCDC designs will allexceed 130C with maximum input voltage in the final application. Furthermore,according to the measured temperatures with the maximum input voltages, the useof single conductors in the windings of DCDC designs will lead to excessive thermalrise in the final application.

The measured thermal rise, of other designs than DCDC34m, is more than 50Cthat was used when calculating the allowed losses in 3.5. This means that the lossesare higher in these components than what was calculated in 3.5. In order to lowerthe thermal rise, a method to reduce the resistance of the windings is to decrease thenumber of turns of windings. This increases the operating flux density and decreasesthe value of inductance. Furthermore, this would be more recommendable thandecreasing the inductance by increasing the length of air gap, because increasing theair gap length will increase fringing flux and therefore increase losses and thermalrise near the air gap.

Based on the measured final temperatures of Table 25 and the cost breakdownsin 3.8, DCDC34m and ACDC29 are chosen as the least expensive but usable com-ponents in the final application. Nevertheless, the cooling effect of the forced con-vection, in the final application, should be verified in the future, if the development

72

of the components should continue.

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5 Summary and conclusions

The aim of this thesis was to design flyback transformers for two already existingflyback converters, emphasizing cost efficiency and reliable operation in the designedcomponents. In addition, the operation and parameters of the components were tobe verified by measurements in order to be convinced of the designed operation.

As a result, four different components were designed for the DC-to-DC converterand two for the AC-to-DC converter in the third chapter of the thesis. The designingprocess and calculations were based on the theory presented in the second chapterof the thesis. Furthermore, the differences of the designs were in the sizes of theselected ferrite cores and in the winding wires. Windings made of single conductorswere used in half of the designs as the windings of the other half were made usingLitz wire.

The costs of the different designs was found to depend on the size of the com-ponent and the DCDC34m was the most expensive of the designs. Moreover, thedifference in the cost to the DCDC34 resulted from the used Litz type TIW, whichwas found as the most expensive material among all designs, if the relative costs ofdifferent materials were compared to the costs of the complete components. On theother hand, Litz type enamelled wire was not more expensive than the enamelledwire of single conductor. In addition, the cost of the DCDC34m, using Litz typeTIW in all windings, was calculated according to the relative costs of TIW. As aresult, the relative cost of this component was approximately 292 % of the cost ofDCDC34m.

The measured mutual inductances of the designs matched well the calculatedvalues. The found small differences may result from tolerances originating fromthe production of the cores. In addition, the values of leakage inductances of theprimaries were found to depend on the distance between primaries and secondariesand also on the overall thickness of the windings. Furthermore, the measured valuesof leakage inductances, in the designed components, were less than 2 % of the mutualinductances.

The results of the measured resistances showed that using Litz type wires inthe windings decreases the effects of AC resistance and therefore the amount ofcopper losses in the components. Furthermore, this was also verified as the lowerthermal rises of the components where Litz wire is used, if compared to the equivalentcomponents using single conductors in the windings. In addition, the amount ofcopper losses was found to be higher than the core losses in the designed components.

All the designs were found to operate close to the designed operating points aspart of the intended flyback converter circuits. However, the switching frequency ofthe DC-to-DC converter was closer to the minimum frequency clamp of the controllerthan what was expected according to the calculations. As a result, the mutualinductances of the DCDC designs should be decreased by choosing a lower numberof turns or by increasing the lengths of air gaps. The former is more recommendedover the latter because increasing the length of the air gap will result in higher AClosses and higher thermal rise in the windings near the air gap. In addition, thecurrents saturating the cores of the flyback transformers were found to be higher

74

than the measured values of the peak currents. However, the measurement wasconducted to the components in a typical room temperature and the saturatingcurrents in the higher operating temperatures should be somewhat lower than themeasured values.

As a conclusion, based on the measured results, the designs selected for theauxiliary power supply of the PV inverter are DCDC34m and ACDC29. Nonetheless,the testing of the components should be continued in the actual final application,in the PV inverter. Moreover, the properties of the selected components should betuned in the next iteration round based on the results of this thesis.

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References

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[2] SMA Solar Technology. SMA solar inverters. Network document. Cited22.7.2014. Available: http://www.sma.de/en/products/solarinverters.

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[3] Remmers, K. Inverter, storage and PV system technology, Industry guide 2013.Berlin, Solarpraxis AG, 2013, 93 p.

[4] IEC 61000-3-12. Limits - Limits for harmonic currents produced by equipmentconnected to public low-voltage systems with input current >16 A and ≤75A per phase. First edition. Geneva, International Electrotechnical Commission.2004. 53 p.

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79

A Appendix

In appendix A, tables from [8] are shown for determining DVC, temporary overvolt-ages and clearance and creepage distances.

The table for determining DVCs is presented in the figure A1.

Figure A1: Table for determining DVC [8, p. 49].

The table for determining insulation voltages is presented in the figure A2.

Figure A2: Table for determining impulse withstand voltage and temporary over-voltage [8, p. 71].

In the figure A2, linear interpolation between rows is forbidden for circuits con-nected to grid but is permitted for PV circuits.

Tables for designing clearance and creepage distances are presented in the fig-ures A3 and A4.

80

Figure A3: Table for designing clearances [8, p. 73].

In the figure A3, linear interpolation is permitted between rows.

Figure A4: Table for designing creepage distances [8, p. 75].

In the figure A4, linear interpolation is permitted between rows.

81

B Appendix

The datasheets of ferrite materials 3C30 from Ferroxcube, DMR40 from DMEGCand TP4 from TDG are presented in Appendix B. The datasheets are provided bythe manufacturers of cores.

Figure B1: Ferroxcube 3C30 Mn-Zn ferrite material. [19, p. 78-79.]

82

Figure B2: DMEGC DMR40 Mn-Zn ferrite material. [23]

83

Figure B3: TDG TP4 Mn-Zn ferrite material. [24, p. 24-25.]

84

C Appendix

Calculations of the designs, using initial and chosen values, are presented in Ap-pendix C.

Turns ratios and inductancesDCDC:First iteration

ton =Dmax

fsw=

0.44

45kHz= 9.78 µs

L =V 2Dt

2onfsw

2(VO + vF )IO=

2402V(9.78µs)2 45kHz

2(24V + 1V)3A

L ≈ 1.65mH

iPpk =VDtonL

=240V ∗ 9.78µs

1.65mH≈ 1.42 A

iP,rms = iPpk

√Dmax

3= 1.42A

√0.44

3= 0.54 A

Second iteration

tres = π√LCDS = π

√1.65mH ∗ 100pF ≈ 1.28 µs

Dres = tresfsw = 1.28µs ∗ 45kHz ≈ 0.06

vdrop ≈ (2 ∗RDS,on +R) ∗ iP,rms = (2 ∗ 6Ω + 1 Ω) ∗ 0.54A ≈ 7 V

NPS =Dmax

1−Dmax −Dres

VD − vdropVO + vF

=0.43

1− 0.43− 0.06∗ 240V − 7V

24 V + 1VNPS ≈ 7.86

ton =NPS(VO + vF )(TS − tres)VD − vdrop +NPS(VO + vF )

=7.86(24V + 1 V)( 1

45kHz− 1.28µs)

240V − 7V + 7.86(24V + 1V)

ton ≈ 9.58 µs

Don = ton ∗ fsw = 9.58µs ∗ 45kHz ≈ 0.43

L =(VD − vdrop)2t2onfsw

2(VO + vF )IO=

(240V − 7V)2(9.58µs)2 45kHz

2(24V + 1 V)3 A

L ≈ 1.49 mH

iPpk =(VD − vdrop)ton

L=

(240V − 7V) ∗ 9.58µs

1.49mH≈ 1.50 A

iP,rms = iPpk

√Dmax

3= 1.50A

√0.43

3≈ 0.57 A

ACDC:First iteration

vrating = VD + vr + vspike + 0.3 ∗ VD = VD +NPS(VO + vF ) + 0.6 ∗ VD

NPS =vrating − VD − 0.6 ∗ VD

VO + vF=

1500 V − 390V − 0.6 ∗ 806V

24V + 1VNPS ≈ 8.42

85

ton =NPS(VO + vF )(TS − tres)VD +NPS(VO + vF )

=8.42(24V + 1 V)( 1

100kHz− 500ns)

390V + 8.42(24V + 1V)ton ≈ 3.33 µs

L =V 2Dt

2onfsw

2(VO + vF )IO=

(390V)2(3.33µs)2100kHz

2(24V + 1V)0.86A

L ≈ 3.92 mH

Second iteration

tres = π√LCDS = π

√3.92mH ∗ 100pF ≈ 1.97 µs

ton =NPS(VO + vF )(TS −Dres)

VD +NPS(VO + vF )=

8.42(24V + 1 V)( 1100kHz

− 1.97µs)

390V + 8.42(24V + 1V)ton ≈ 2.81 µs

Don = ton ∗ fsw = 2.81µs ∗ 100kHz ≈ 0.28

L =V 2Dt

2onfsw

2(VO + vF )IO=

(390V)2(2.81µs)2100kHz

2(24 V + 1 V)0.86A≈ 2.79 mH

iPpk =VDtonL

=390V ∗ 2.81µs

2.79mH≈ 0.39 A

iP,rms = iPpk

√tonfsw

3= 0.39A

√2.81µs 100kHz

3= 0.12 A

vdrop ≈ (RDS,on +R) ∗ iP,rms = (6Ω + 1Ω) ∗ 0.12A = 0.84 V

Initial energy throughput of the flyback transformersDCDC:

E = i2PpkL = (1.50A)2 ∗ 1.49mH ≈ 3.35 mJ

ACDC:

E = i2PpkL = (0.39A)2 ∗ 2.79mH =≈ 0.42 mJ

Number of turns, air gap length and actual mutual inductanceDCDC34:

N =(VD − vdrop)ton

∆B ∗ Ac=

(240V − 7V)9.58µs

0.25T ∗ 0.971cm2≈ 91.6

NAUX =NP

N=

vrvAUX

=7.67 ∗ 25V

19V≈ 10.1

N =NP

NAUX

=92

10.1≈ 9.1

NAUX =92

9≈ 10.2

LM = FL =

(1 +

lg√Ac

ln2G

lg

)µ0N

2Ac ∗ 10−2

lg + lcµi

LM =

(1 +

0.1cm√0.971cm2

ln2 ∗ 2.36cm

0.1cm

)1.26H/m ∗ 922 ∗ 0.971cm2 ∗ 10−8

0.1cm + 7.86cm2300

86

LM ≈ 1.39 mH

fsw =(VD − vdrop)2D2

on

2 ∗ (VO + vF ) ∗ IO ∗ L=

(240V − 7V)2 ∗ 0.432

2 ∗ (24V + 1V) ∗ 3A ∗ 1.39mH≈ 48 kHz

tres = π√LCDS = π

√1.39mH ∗ 100pF ≈ 1.17 µs

Dres = tresfsw = 1.17µs ∗ 48kHz ≈ 0.06

ton =NPS(VO + vF )(TS −Dres)

VD − vdrop +NPS(VO + vF )=

7.67(24V + 1 V)( 148kHz

− 1.17µs)

240V − 7V + 7.86(24V + 1V)

ton ≈ 8.88 µs

Don = ton ∗ fsw = 8.88µs ∗ 48kHz ≈ 0.43

iPpk =(VD − vdrop)ton

L=

(240V − 7V) ∗ 8.88µs

1.39mH≈ 1.49 A

iP,rms = iPpk

√Don

3= 1.49A

√0.43

3= 0.56 A

∆B =(VD − vdrop)ton

NP ∗ Ac=

(240V − 7V)8.88µs

92 ∗ 0.971cm2≈ 0.23 T

DCDC29:

N =(VD − vdrop)ton

∆B ∗ Ac=

(240V − 7V)9.58µs

0.25T ∗ 0.768cm2≈ 116.2

LM = FL =

(1 +

lg√Ac

ln2G

lg

)µ0N

2Ac ∗ 10−2

lg + lcµi

LM =

(1 +

0.08cm√0.768cm2

ln2 ∗ 2.2cm

0.08cm

)1.26H/m ∗ 922 ∗ 0.768cm2 ∗ 10−8

0.08cm + 7.2cm2300

LM ≈ 1.39 mH

fsw =(VD − vdrop)2D2

on

2 ∗ (VO + vF ) ∗ IO ∗ L=

(240V − 7V)2 ∗ 0.432

2 ∗ (24V + 1V) ∗ 3A ∗ 1.35mH≈ 50 kHz

tres = π√LCDS = π

√1.35mH ∗ 100pF ≈ 1.15 µs

Dres = tresfsw = 1.15µs ∗ 50kHz ≈ 0.06

ton =NPS(VO + vF )(TS −Dres)

VD − vdrop +NPS(VO + vF )=

7.67(24V + 1 V)( 150kHz

− 1.15µs)

240V − 7V + 7.67(24V + 1V)

ton ≈ 8.51 µs

Don = ton ∗ fsw = 8.51µs ∗ 50kHz ≈ 0.43

iPpk =(VD − vdrop)ton

L=

(240V − 7V) ∗ 8.51µs

1.35mH≈ 1.47 A

iP,rms = iPpk

√Don

3= 1.47A

√0.43

3= 0.56 A

∆B =(VD − vdrop)ton

NP ∗ Ac=

(240V − 7V)8.51µs

92 ∗ 0.768cm2≈ 0.28 T

ACDC:

N =VDton

∆B ∗ Ac=

390V ∗ 2.81µs

0.25T ∗ 0.768cm2≈ 57.1

87

LM = FL =

(1 +

lg√Ac

ln2G

lg

)µ0N

2Ac ∗ 10−2

lg + lcµi

LM =

(1 +

0.03cm√0.768cm2

ln2 ∗ 2.2cm

0.03cm

)1.26H/m ∗ 572 ∗ 0.768cm2 ∗ 10−8

0.03cm + 7.2cm2300

LM ≈ 1.11 mH

LM =

(1 +

0.03cm√0.768cm2

ln2 ∗ 2.2cm

0.03cm

)1.26H/m ∗ 882 ∗ 0.768cm2 ∗ 10−8

0.03cm + 7.2cm2300

LM ≈ 2.65 mH

NAUX =NP

N=

vrvAUX

=8 ∗ 25V

19V≈ 10.5

N =NP

NAUX

=88

10.5≈ 8.4

NAUX =88

8= 11

VAUX ≈ vrNAUX

=8 ∗ 25V

11≈ 18.2 V

tres = π√LCDS = π

√2.65mH ∗ 100pF ≈ 1.62 µs

ton =NPS(VO + vF )(TS −Dres)

VD +NPS(VO + vF )=

8(24V + 1 V)( 1100kHz

− 1.62µs)

390V + 8(24V + 1V)ton ≈ 2.84 µs

Don = ton ∗ fsw = 2.84µs ∗ 100kHz ≈ 0.29

iPpk =VDtonL

=390V ∗ 2.84µs

2.65mH≈ 0.42 A

iP,rms = iPpk

√Don

3= 0.42A

√0.29

3= 0.13 A

∆B =(VD − vdrop)ton

NP ∗ Ac=

390V2.84µs

88 ∗ 0.768cm2≈ 0.16 T

Cooling surface area and allowed lossesETD34:

A1 ≈ 2X(W + Z) = 2 ∗ 1.1cm ∗ (3.5cm + 3.46cm) ≈ 15.3 cm2

A2 ≈ 4YW = 4 ∗ 0.55cm ∗ 3.5cm ≈ 7.7 cm2

A3 ≈ 4Y (Z − 2Y ) = 4 ∗ 0.55cm(3.46cm− 2 ∗ 0.55cm) ≈ 5.2 cm2

A4 ≈ 2π

(W − 2Y

2

)(Z − 2Y )− 2X(Z − 2Y )

A4 ≈ 2π

(3.5cm− 2 ∗ 0.55cm

2

)(3.46cm− 2 ∗ 0.55cm)−

2 ∗ 1.1cm(3.46cm− 2 ∗ 0.55cm) ≈ 12.6 cm2

A5 ≈ 2π

(W − 2Y

2

)2

− 2X(W − 2Y )

88

A5 ≈ 2π

(3.5cm− 2 ∗ 0.55cm

2

)2

− 2 ∗ 1.1cm(3.5cm− 2 ∗ 0.55cm) ≈ 3.8 cm2

A = A1 + A2 + A3 + A4 + A5

A ≈ 15.3cm2 + 7.7cm2 + 5.2cm2 + 12.6cm2 + 3.8cm2 = 44.6 cm2

Ploss = ∆T 1.1A = 50C1.144.6cm2 ≈ 3298 mW

ETD29:

A1 ≈ 2X(W + Z) = 2 ∗ 0.98cm ∗ (3.06cm + 3.16cm) ≈ 12.2 cm2

A2 ≈ 4YW = 4 ∗ 0.50cm ∗ 3.06cm ≈ 6.1 cm2

A3 ≈ 4Y (Z − 2Y ) = 4 ∗ 0.50cm(3.16cm− 2 ∗ 0.50cm) ≈ 4.3cm2

A4 ≈ 2π

(W − 2Y

2

)(Z − 2Y )− 2X(Z − 2Y )

A4 ≈ 2π

(3.06cm− 2 ∗ 0.50cm

2

)(3.16cm− 2 ∗ 0.50cm)−

2 ∗ 0.98cm(3.16cm− 2 ∗ 0.50cm) ≈ 9.77 cm2

A5 ≈ 2π

(W − 2Y

2

)2

− 2X(W − 2Y )

A5 ≈ 2π

(3.06cm− 2 ∗ 0.50cm

2

)2

− 2 ∗ 0.98cm(3.06cm− 2 ∗ 0.50cm) ≈ 2.6 cm2

A = A1 + A2 + A3 + A4 + A5

A ≈ 12.2cm2 + 6.1cm2 + 4.3cm2 + 9.77cm2 + 2.6cm2 = 35.0 cm2

Ploss = ∆T 1.1A = 50C1.135.0cm2 ≈ 2588 mW

Calculated flux densities, core losses and copper lossesDCDC34:

∆B =(VD − vdrop)ton

NPAc=

(240V − 7V)8.88µs

92 ∗ 0.971cm2≈ 0.23 T

Ve = Lc ∗ Ac = 7.86cm ∗ 0.971cm2 ≈ 7.63 cm3

PFe = ρFe ∗ Ve = 24W/cm3 ∗ 7.63cm3 ≈ 180 mW

PCu = Ploss − PFe = 3298mW − 180mW ≈ 3.1 W

DCDC29:

∆B =(VD − vdrop)ton

NPAc=

(240V − 7V)8.51µs

92 ∗ 0.768cm2≈ 0.28 T

Ve = Lc ∗ Ac = 7.20cm ∗ 0.768cm2 ≈ 5.53 cm3

PFe = ρFe ∗ Ve = 24W/cm3 ∗ 5.53cm3 ≈ 240 mW

PCu = Ploss − PFe = 2588mW − 240mW ≈ 2.3 W

ACDC29:

∆B =VDtonNPAc

=390V ∗ 8.51µs

88 ∗ 0.768cm2≈ 0.16 T

89

Ve = Lc ∗ Ac = 7.20cm ∗ 0.768cm2 ≈ 5.53 cm3

PFe = ρFe ∗ Ve = 32W/cm3 ∗ 5.53cm3 ≈ 180 mW

PCu = Ploss − PFe = 2588mW − 180mW ≈ 2.4 W

90

Clearance and creepage distancesInterpolation of the impulse withstand voltage:From figure A2

tan(θ) =6000V − 4000V

1500V − 849Vvimpulse = 4000V + (1100V − 849V)tan(θ)

= 4000V + (1100V − 849V)6000V − 4000V

1500V − 849V≈ 4771 V

Interpolation of the clearance:From figure A3

tan(θ) =5.5mm− 3mm

6000V − 4000VClearance = 3mm + (4771V − 4000V)tan(θ)

= 3mm + 771V5.5mm− 3mm

6000V − 4000V≈ 4 mm

Interpolation of the creepage:From figure A4

tanθ =6.3mm− 5.0mm

1250V − 1000VCreepage = 5.0mm + (1100V − 1000V)tan(θ)

= 5mm + 100V6.3mm− 5.0mm

1250V − 1000V≈ 5.5 mm

Rms values of currentsDCDC34:

iP,rms = iPpk

√Don

3= 1.49A

√8.88µs ∗ 48kHz

3= 0.57 A

toff =1

48kHz− 8.88µs− 1.17µs ≈ 10.8 µs

iSpk = N2PS

(VO + vF )toffL

= 7.672 (24V + 1V)10.8µs

1.39mH≈ 11.4 A

iS,rms = iSpk

√Doff

3= 11.4A

√10.8µs ∗ 48kHz

3≈ 4.7 A

iS1,rms =IO1

IO1 + IO2

iS,rms =1.8A

1.8A + 1.2A4.7A ≈ 2.8 A

iS2,rms = iS,rms − iS1,rms = 4.7A− 2.8A ≈ 1.9 A

itot = iP,rms +NS

NP

iS,rms = 0.57A +12

924.7A ≈ 1.18 A

91

DCDC29:

iP,rms = iPpk

√Don

3= 1.49A

√8.51µs ∗ 50kHz

3= 0.57 A

toff =1

50kHz− 8.51µs− 1.15µs ≈ 10.3 µs

iSpk = N2PS

(VO + vF )toffL

= 7.672 (24V + 1V)10.3µs

1.35mH≈ 11.2 A

iS,rms = iSpk

√Doff

3= 11.2A

√10.3µs ∗ 50kHz

3≈ 4.6 A

iS1,rms =IO1

IO1 + IO2

iS,rms =1.8A

1.8A + 1.2A4.6A ≈ 2.8 A

iS2,rms = iS,rms − iS1,rms = 4.6A− 2.8A ≈ 1.8 A

itot = iP,rms +NS

NP

iS,rms = 0.56A +12

924.6A ≈ 1.16 A

ACDC29:

iP,rms = iPpk

√Don

3= 0.42A

√2.84µs ∗ 100kHz

3= 0.13 A

toff =1

100kHz− 2.84µs− 1.62µs ≈ 5.54 µs

iSpk = N2PS

(VO + vF )toffL

= 82 (24V + 1V)5.54µs

2.65mH≈ 3.34 A

iS,rms = iSpk

√Doff

3= 3.34A

√5.54µs ∗ 100kHz

3≈ 1.44 A

iS1,rms =IO1

IO1 + IO2

iS,rms =0.41A

0.41A + 0.45A1.44A ≈ 0.69 A

iS2,rms = iS,rms − iS1,rms = 1.44A− 0.69A ≈ 0.75 A

itot = iP,rms +NS

NP

iS,rms = 0.13A +11

881.44A ≈ 0.31 A

Cross sectional areas and maximum diameters of winding wiresDCDC34:

αP =NP iP,rmsNP itot

=0.57A

1.2A≈ 0.5

αS1 =NSiS1,rms

NP itot=

12 ∗ 2.8A

92 ∗ 1.2A≈ 0.3

αS2 =NSiS2,rms

NP itot=

12 ∗ 1.9A

92 ∗ 1.2A≈ 0.2

AwP ≤ αPKuWA

NP

=0.5 ∗ 0.33 ∗ 123mm2

92≈ 0.22 mm2

dwP ≤ 2

√AwPπ

= 2

√0.22mm2

π≈ 0.53 mm

92

AwS1 ≤αS1KuWA

NS

=0.3 ∗ 0.33 ∗ 123mm2

12≈ 1.0 mm2

dwS1 ≤ 2

√AwS1

π= 2

√1.0mm2

π≈ 1.13 mm

AwS2 ≤αS2KuWA

NS

=0.2 ∗ 0.33 ∗ 123mm2

12≈ 0.68 mm2

dwS2 ≤ 2

√AwS2

π= 2

√0.68mm2

π≈ 0.93 mm

DCDC29:

αP =NP iP,rmsNP itot

=0.57A

1.2A≈ 0.48

αS1 =NSiS1,rms

NP itot=

12 ∗ 2.8A

92 ∗ 1.2A≈ 0.3

αS2 =NSiS2,rms

NP itot=

12 ∗ 1.9A

92 ∗ 1.2A≈ 0.2

AwP ≤ αPKuWA

NP

=0.48 ∗ 0.33 ∗ 90mm2

92≈ 0.15 mm2

dwP ≤ 2

√AwPπ

= 2

√0.15mm2

π≈ 0.44 mm

AwS1 ≤αS1KuWA

NS

=0.3 ∗ 0.33 ∗ 90mm2

12≈ 0.74 mm2

dwS1 ≤ 2

√AwS1

π= 2

√0.74mm2

π≈ 0.97 mm

AwS2 ≤αS2KuWA

NS

=0.2 ∗ 0.33 ∗ 90mm2

12≈ 0.49 mm2

dwS2 ≤ 2

√AwS2

π= 2

√0.49mm2

π≈ 0.79 mm

ACDC29:

αP =NP iP,rmsNP itot

=0.13A

0.31A≈ 0.42

αS1 =NSiS1,rms

NP itot=

11 ∗ 0.69A

88 ∗ 0.31A≈ 0.28

αS2 =NSiS2,rms

NP itot=

11 ∗ 0.75A

88 ∗ 0.31A≈ 0.30

AwP ≤ αPKuWA

NP

=0.42 ∗ 0.33 ∗ 90mm2

88≈ 0.14 mm2

dwP ≤ 2

√AwPπ

= 2

√0.14mm2

π≈ 0.42 mm

AwS1 ≤αS1KuWA

NS

=0.28 ∗ 0.33 ∗ 90mm2

11≈ 0.76 mm2

93

dwS1 ≤ 2

√AwS1

π= 2

√0.76mm2

π≈ 0.98 mm

AwS2 ≤αS2KuWA

NS

=0.3 ∗ 0.33 ∗ 90mm2

11≈ 0.81 mm2

dwS2 ≤ 2

√AwS2

π= 2

√0.81mm2

π≈ 1.0 mm

Skin depth and numbers of strandsSkin depth:

δ =

√ρc

πfµ0

=

√2.3 ∗ 10−8Ωm

π ∗ 160kHz ∗ 1.26 ∗ 10−6≈ 0.19 mm

dwP = 2 ∗ δ = 2 ∗ 0.19mm = 0.38 mm

DCDC34m:

dwi,p =dwi√p

p =d2wP

d2wP,p

=(0.53mm)2

(0.25mm)2≈ 4

p =d2wS1

d2wS1,p

=(1.13mm)2

(0.3mm)2≈ 14

p =d2wS2

d2wS2,p

=(0.93mm)2

(0.3mm)2≈ 10

DCDC29m:

p =d2wP

d2wP,p

=(0.44mm)2

(0.2mm)2≈ 5

p =d2wS1

d2wS1,p

=(0.97mm)2

(0.28mm)2≈ 12

p =d2wS2

d2wS2,p

=(0.79mm)2

(0.23mm)2≈ 12

ACDC29m:

p =d2wP

d2wP,p

=(0.42mm)2

(0.2mm)2≈ 4

p =d2wS1

d2wS1,p

=(0.98mm)2

(0.28mm)2≈ 12

p =d2wS2

d2wS2,p

=(1.0mm)2

(0.27mm)2≈ 14

94

Ohmic resistances and losses

Figure C1: Distances for determining radius r to calculate mean lengths per turn.DCDC34 on the left and DCDC29 on the right.

Distances from Figure C1.DCDC34:

rP =dc2

+dwP,p

2+

4.9mm

2=

13.6mm

2+

0.55

2+

4.9mm

2≈ 9.5 mm

rS1 =dc2

+ 1.9mm =13.6mm

2+ 1.9mm ≈ 8.7 mm

rS2 =dc2

+ 3.2mm =13.6mm

2+ 3.2mm ≈ 10 mm

MLTP = 2πrP = 2 ∗ π ∗ 9.5mm ≈ 59.7 mm

MLTS1 = 2πrS1 = 2 ∗ π ∗ 8.7mm ≈ 54.7 mm

MLTS2 = 2πrS2 = 2 ∗ π ∗ 10mm ≈ 62.6 mm

DCDC29 and ACDC29:

rP =dc2

+dwP,p

2+

4.2mm

2=

12.2mm

2+

0.4

2+

4.2mm

2≈ 8.4 mm

rS1 =dc2

+ 1.6mm =12.2mm

2+ 1.6mm ≈ 7.7 mm

rS2 =dc2

+ 2.7mm =12.2mm

2+ 2.7mm ≈ 8.8 mm

MLTP = 2πrP = 2 ∗ π ∗ 8.4mm ≈ 52.8 mm

MLTS1 = 2πrS1 = 2 ∗ π ∗ 7.7mm ≈ 48.4 mm

MLTS2 = 2πrS2 = 2 ∗ π ∗ 8.8mm ≈ 55.3 mm

Resistances and lossesDCDC34:

R0P = ρc4MLTPNP

πd2wP,p

= 2.3 ∗ 10−8Ωm4 ∗ 59.7 ∗ 10−3m ∗ 92

π ∗ (0.55 ∗ 10−3m)2≈ 0.53 Ω

R0S1 = ρc4MLTS1NS

πd2wS1,p

= 2.3 ∗ 10−8Ωm4 ∗ 54.7 ∗ 10−3m ∗ 12

π ∗ (1 ∗ 10−3m)2≈ 0.019 Ω

95

R0S2 = ρc4MLTS2NS

πd2wS2,p

= 2.3 ∗ 10−8Ωm4 ∗ 62.6 ∗ 10−3m ∗ 12

π ∗ (0.9 ∗ 10−3m)2≈ 0.027 Ω

JP =iP,rms

pP ∗ π dwP,p

2

2 =0.57A

π 0.055cm2

2 ≈ 240 A/cm2

JS1 =iS1,rms

pS1 ∗ π dwS1,p

2

2 =2.8A

π 0.1cm2

2 ≈ 357 A/cm2

JS2 =iS2,rms

pS2 ∗ π dwS2,p

2

2 =1.9A

π 0.09cm2

2 ≈ 299 A/cm2

PCu0 = R0P ∗ i2P,rms +R0S1 ∗ i2S1,rms +R0S2 ∗ i2S2,rms

PCu0 = 0.53Ω ∗ (0.57A)2 + 0.019Ω ∗ (2.8A)2 + 0.027Ω ∗ (1.9A)2 ≈ 0.42 W

DCDC34m:

R0P = ρc4MLTPNP

πd2wP,p

= 2.3 ∗ 10−8Ωm4 ∗ 59.7 ∗ 10−3m ∗ 92

π ∗ (√

3 ∗ 0.25 ∗ 10−3m)2≈ 0.86 Ω

R0S1 = ρc4MLTS1NS

πd2wS1,p

= 2.3 ∗ 10−8Ωm4 ∗ 54.7 ∗ 10−3m ∗ 12

π ∗ (√

8 ∗ 0.30 ∗ 10−3m)2≈ 0.027 Ω

R0S2 = ρc4MLTS2NS

πd2wS2,p

= 2.3 ∗ 10−8Ωm4 ∗ 62.6 ∗ 10−3m ∗ 12

π ∗ (√

7 ∗ 0.3 ∗ 10−3m)2≈ 0.035 Ω

JP =iP,rms

pP ∗ π dwP,p

2

2 =0.57A

3 ∗ π 0.025cm2

2 ≈ 387 A/cm2

JS1 =iS1,rms

pS1 ∗ π dwS1,p

2

2 =2.8A

8 ∗ π 0.03cm2

2 ≈ 495 A/cm2

JS2 =iS2,rms

pS2 ∗ π dwS2,p

2

2 =1.9A

7 ∗ π 0.03cm2

2 ≈ 384 A/cm2

PCu0 = R0P ∗ i2P,rms +R0S1 ∗ i2S1,rms +R0S2 ∗ i2S2,rms

PCu0 = 0.86Ω ∗ (0.57A)2 + 0.027Ω ∗ (2.8A)2 + 0.035Ω ∗ (1.9A)2 ≈ 0.62 W

DCDC29:

R0P = ρc4MLTPNP

πd2wP,p

= 2.3 ∗ 10−8Ωm4 ∗ 52.8 ∗ 10−3m ∗ 92

π ∗ (0.4 ∗ 10−3m)2≈ 0.89 Ω

R0S1 = ρc4MLTS1NS

πd2wS1,p

= 2.3 ∗ 10−8Ωm4 ∗ 48.4 ∗ 10−3m ∗ 12

π ∗ (0.9 ∗ 10−3m)2≈ 0.021 Ω

R0S2 = ρc4MLTS2NS

πd2wS2,p

= 2.3 ∗ 10−8Ωm4 ∗ 55.3 ∗ 10−3m ∗ 12

π ∗ (0.7 ∗ 10−3m)2≈ 0.040 Ω

JP =iP,rms

pP ∗ π dwP,p

2

2 =0.57A

π 0.04cm2

2 ≈ 454 A/cm2

JS1 =iS1,rms

pS1 ∗ π dwS1,p

2

2 =2.8A

π 0.09cm2

2 ≈ 440 A/cm2

96

JS2 =iS2,rms

pS2 ∗ π dwS2,p

2

2 =1.8A

π 0.07cm2

2 ≈ 468 A/cm2

PCu0 = R0P ∗ i2P,rms +R0S1 ∗ i2S1,rms +R0S2 ∗ i2S2,rms

PCu0 = 0.89Ω ∗ (0.57A)2 + 0.021Ω ∗ (2.8A)2 + 0.040Ω ∗ (1.8A)2 ≈ 0.58 W

DCDC29m:

R0P = ρc4MLTPNP

πd2wP,p

= 2.3 ∗ 10−8Ωm4 ∗ 52.8 ∗ 10−3m ∗ 92

π ∗ (√

3 ∗ 0.20 ∗ 10−3m)2≈ 1.19 Ω

R0S1 = ρc4MLTS1NS

πd2wS1,p

= 2.3 ∗ 10−8Ωm4 ∗ 48.4 ∗ 10−3m ∗ 12

π ∗ (√

7 ∗ 0.28 ∗ 10−3m)2≈ 0.031 Ω

R0S2 = ρc4MLTS2NS

πd2wS2,p

= 2.3 ∗ 10−8Ωm4 ∗ 55.3 ∗ 10−3m ∗ 12

π ∗ (√

7 ∗ 0.23 ∗ 10−3m)2≈ 0.052 Ω

JP =iP,rms

pP ∗ π dwP,p

2

2 =0.57A

3 ∗ π 0.02cm2

2 ≈ 605 A/cm2

JS1 =iS1,rms

pS1 ∗ π dwS1,p

2

2 =2.8A

7 ∗ π 0.028cm2

2 ≈ 650 A/cm2

JS2 =iS2,rms

pS2 ∗ π dwS2,p

2

2 =1.8A

7 ∗ π 0.023cm2

2 ≈ 619 A/cm2

PCu0 = R0P ∗ i2P,rms +R0S1 ∗ i2S1,rms +R0S2 ∗ i2S2,rms

PCu0 = 1.19Ω ∗ (0.57A)2 + 0.031Ω ∗ (2.8A)2 + 0.052Ω ∗ (1.8A)2 ≈ 0.80 W

ACDC29:

R0P = ρc4MLTPNP

πd2wP,p

= 2.3 ∗ 10−8Ωm4 ∗ 52.8 ∗ 10−3m ∗ 88

π ∗ (0.4 ∗ 10−3m)2≈ 0.85 Ω

R0S1 = ρc4MLTS1NS

πd2wS1,p

= 2.3 ∗ 10−8Ωm4 ∗ 48.4 ∗ 10−3m ∗ 11

π ∗ (0.8 ∗ 10−3m)2≈ 0.024 Ω

R0S2 = ρc4MLTS2NS

πd2wS2,p

= 2.3 ∗ 10−8Ωm4 ∗ 55.3 ∗ 10−3m ∗ 11

π ∗ (0.9 ∗ 10−3m)2≈ 0.022 Ω

JP =iP,rms

pP ∗ π dwP,p

2

2 =0.13A

π 0.04cm2

2 ≈ 103 A/cm2

JS1 =iS1,rms

pS1 ∗ π dwS1,p

2

2 =0.69A

π 0.08cm2

2 ≈ 137 A/cm2

JS2 =iS2,rms

pS2 ∗ π dwS2,p

2

2 =0.75A

π 0.09cm2

2 ≈ 118 A/cm2

PCu0 = R0P ∗ i2P,rms +R0S1 ∗ i2S1,rms +R0S2 ∗ i2S2,rms

PCu0 = 0.85Ω ∗ (0.13A)2 + 0.024Ω ∗ (0.69A)2 + 0.022Ω ∗ (0.75A)2 ≈ 0.04 W

97

ACDC29m:

R0P = ρc4MLTPNP

πd2wP,p

= 2.3 ∗ 10−8Ωm4 ∗ 52.8 ∗ 10−3m ∗ 88

π ∗ (√

3 ∗ 0.20 ∗ 10−3m)2≈ 1.13 Ω

R0S1 = ρc4MLTS1NS

πd2wS1,p

= 2.3 ∗ 10−8Ωm4 ∗ 48.4 ∗ 10−3m ∗ 11

π ∗ (√

6 ∗ 0.28 ∗ 10−3m)2≈ 0.033 Ω

R0S2 = ρc4MLTS2NS

πd2wS2,p

= 2.3 ∗ 10−8Ωm4 ∗ 55.3 ∗ 10−3m ∗ 11

π ∗ (√

7 ∗ 0.27 ∗ 10−3m)2≈ 0.035 Ω

JP =iP,rms

pP ∗ π dwP,p

2

2 =0.13A

3 ∗ π 0.02cm2

2 ≈ 138 A/cm2

JS1 =iS1,rms

pS1 ∗ π dwS1,p

2

2 =0.69A

6 ∗ π 0.028cm2

2 ≈ 187 A/cm2

JS2 =iS2,rms

pS2 ∗ π dwS2,p

2

2 =0.75A

7 ∗ π 0.027cm2

2 ≈ 187 A/cm2

PCu0 = R0P ∗ i2P,rms +R0S1 ∗ i2S1,rms +R0S2 ∗ i2S2,rms

PCu0 = 1.13Ω ∗ (0.13A)2 + 0.033Ω ∗ (0.69A)2 + 0.035Ω ∗ (0.75A)2 ≈ 0.05 W

Costs of DCDC34m using TIW in all windings.

Primary = 0.77% ∗ 28.7% ∗ 59.7mm

55.3mm∗ 7.67 ≈ 183 %

Secondaries = 32.4% + 32.4% ∗ 54.7mm

62.6mm≈ 60.7 %

Labor =2

3∗ 19.9% ≈ 13.3 %

Other =9

10∗ 8.2% ≈ 7.4 %

Complete = Core+ Coilformer + Primary + Secondaries+ Labor +

Other

Complete = 19.0% + 8.2% + 183% + 60.7% + 13.3% + 7.4% ≈ 292 %